Set superpartitions and superspace duality modules

2.1 Alternants in superspace

Recall that if V is an ${\mathfrak {S}}_n$-module, a vector $v \in V$ is an alternant if

(2.1)$$ \begin{align} w \cdot v = {\mathrm {sign}}(w) \cdot v \quad \quad \text{for all } w \in {\mathfrak{S}}_n. \end{align} $$

The superspace Vandermondes $\delta _{n,k} \in \Omega _n$ are alternants used to construct the quotient rings ${\mathbb {W}}_{n,k}$. In order to place ${\mathbb {W}}_{n,k}$ in the proper inductive context, we will need a more general family of alternants and rings.

Definition 2.1. Let $n, k, s \geq 0$ be integers. Define $\delta _{n,k,s} \in \Omega _n$ to be the element

(2.2)$$ \begin{align} \delta_{n,k,s} = \varepsilon_n \cdot ( x_1^{k-1} \cdots x_{n-s}^{k-1} x_{n-s+1}^{s-1} \cdots x_{n-1}^1 x_n^0 \times \theta_1 \cdots \theta_{n-k}) , \end{align} $$

where $\varepsilon _n = \sum _{w \in {\mathfrak {S}}_n} {\mathrm {sign}}(w) \cdot w \in {\mathbb {Q}}[{\mathfrak {S}}_n]$. Let ${\mathrm {ann}} \, \delta _{n,k,s} \subseteq \Omega _n$ be the annihilator of $\delta _{n,k,s}$, and define ${\mathbb {W}}_{n,k,s}$ to be the quotient ring

(2.3)$$ \begin{align} {\mathbb{W}}_{n,k,s} = \Omega_n / {\mathrm {ann}} \, \delta_{n,k,s}. \end{align} $$

By convention, if $n < k$, then $\delta _{n,k,s} = 0$. In the special case $s = k$, we have ${\mathbb {W}}_{n,k,k} = {\mathbb {W}}_{n,k}$. We only use the ring ${\mathbb {W}}_{n,k,s}$ in the range $k \geq s$.

2.2 Harmonics in superspace

It will be convenient to have a model for ${\mathbb {W}}_{n,k,s}$ as a subspace rather than a quotient of $\Omega _n$. To this end, we define the harmonic module ${\mathbb {H}}_{n,k,s} \subseteq \Omega _n$ as follows.

More explicitly, the harmonic module ${\mathbb {H}}_{n,k,s}$ is the smallest linear subspace of $\Omega _n$ containing $\delta _{n,k,s}$, which is closed under the action of the commuting partial derivatives $\partial /\partial x_1, \dots , \partial /\partial x_n$ as well as the anticommuting partial derivatives $\partial /\partial \theta _1, \dots , \partial /\partial \theta _n$. The subspace ${\mathbb {H}}_{n,k,s}$ is a bigraded ${\mathfrak {S}}_n$-module.

We have a natural inclusion map ${\mathbb {H}}_{n,k,s} \hookrightarrow \Omega _n$. The composition

(2.4)$$ \begin{align} {\mathbb{H}}_{n,k,s} \hookrightarrow \Omega_n \twoheadrightarrow {\mathbb{W}}_{n,k,s} \end{align} $$

of this inclusion with the canonical projection of $\Omega _n$ onto ${\mathbb {W}}_{n,k,s}$ is an isomorphism of bigraded ${\mathfrak {S}}_n$-modules. We make use of ${\mathbb {H}}_{n,k,s}$ when we need to consider superspace elements in $\Omega _n$ rather than cosets in ${\mathbb {W}}_{n,k,s}$.

2.3 Symmetric functions and ${\mathfrak {S}}_n$-modules

Throughout this paper, we use the following standard q-analogs of numbers, factorials and binomial coefficients:

(2.5)$$ \begin{align} [n]_q = \frac{q^n - 1}{q - 1} = 1 + q + \cdots + q^{n-1}, \quad [n]!_q = [n]_q [n-1]_q \cdots [1]_q, \quad {n \brack k}_q = \frac{[n]!_q}{[k]!_q \cdot [n-k]!_q}.\\[-16pt]\nonumber \end{align} $$

A partition $\lambda $ of $ n$ is a weakly decreasing sequence $\lambda = (\lambda _1 \geq \cdots \geq \lambda _k)$ of positive integers which sum to n. We write $\lambda \vdash n$ to mean that $\lambda $ is a partition of n.

Let $\Lambda = \bigoplus _{n \geq 0} \Lambda _n$ be the ring of symmetric functions in an infinite variable set ${\mathbf {x}} = (x_1, x_2, \dots )$ over the ground field ${\mathbb {Q}}(q,z)$. Bases of the $n^{th}$ graded piece $\Lambda _n$ of this ring are indexed by partitions $\lambda \vdash n$. We let

(2.6)$$ \begin{align} \{ m_{\lambda} \,:\, \lambda \vdash n \}, \quad \{ e_{\lambda} \,:\, \lambda \vdash n \}, \quad \{ h_{\lambda} \,:\, \lambda \vdash n\}\quad \text{and} \quad \{ s_{\lambda} \,:\, \lambda \vdash n\}\\[-16pt]\nonumber \end{align} $$

be the monomial, elementary, homogeneous and Schur bases of $\Lambda _n$. Given two partitions $\lambda , \mu $ with $\lambda _i \geq \mu _i$ for all i, we let $s_{\lambda /\mu }$ be the corresponding skew Schur function.

A formal power series F in the variable set ${\mathbf {x}} = (x_1, x_2, \dots )$ of bounded degree is quasisymmetric if the coefficient of $x_1^{a_1} \cdots x_n^{a_n}$ equals the coefficient of $x_{i_1}^{a_1} \cdots x_{i_n}^{a_n}$ for any strictly increasing sequence $i_1 < \cdots < i_n$ of indices. Given a subset $S \subseteq [n-1]$, the fundamental quasisymmetric function of degree n is

(2.7)$$ \begin{align} F_{S,n} = \sum_{\substack{i_1 \leq \cdots \leq i_n \\ j \in S \, \Rightarrow \, i_j < i_{j+1}}} x_{i_1} \cdots x_{i_n}.\\[-16pt]\nonumber \end{align} $$

We will encounter the formal power series $F_{S,n}$ exclusively in the case where S is the inverse descent set of a permutation $w \in {\mathfrak {S}}_n$. This is the set

(2.8)$$ \begin{align} {\mathrm {iDes}}(w) = \{ 1 \leq i \leq n-1 \, : \, w^{-1}(i)> w^{-1}(i+1) \}.\\[-16pt]\nonumber \end{align} $$

We let $\langle -, - \rangle $ be the Hall inner product on $\Lambda _n$ obtained by declaring the Schur functions $s_{\lambda }$ to be orthonormal. For any $F \in \Lambda $, we have a ‘skewing’ operator $F^{\perp }: \Lambda \rightarrow \Lambda $ characterised by

(2.9)$$ \begin{align} \langle F^{\perp} G, H \rangle = \langle G, F H \rangle\\[-16pt]\nonumber \end{align} $$

for all $G, H \in \Lambda $. We will make use of the following fact.

Lemma 2.3 follows from the fact that either of the sets $\{e_1, e_2, \dots \}$ or $\{h_1, h_2, \dots \}$ are algebraically independent generating sets of the ring $\Lambda $ of symmetric functions.

Irreducible representations of ${\mathfrak {S}}_n$ are in bijective correspondence with partitions $\lambda $ of n. If $\lambda \vdash n$ is a partition, let $S^{\lambda }$ be the corresponding irreducible ${\mathfrak {S}}_n$-module. If V is any finite-dimensional ${\mathfrak {S}}_n$-module, there are unique multiplicities $c_{\lambda } \geq 0$, such that $V \cong \bigoplus _{\lambda \vdash n} c_{\lambda } S^{\lambda }$. The Frobenius image of V is the symmetric function

(2.10)$$ \begin{align} {\mathrm {Frob}}(V) = \sum_{\lambda \vdash n} c_{\lambda} s_{\lambda} \in \Lambda_n\\[-16pt]\nonumber \end{align} $$

obtained by replacing each irreducible $S^{\lambda }$ with the corresponding Schur function $s_{\lambda }$.

Given two positive integers $n, m$, we have the corresponding parabolic subgroup ${\mathfrak {S}}_n \times {\mathfrak {S}}_m \subseteq {\mathfrak {S}}_{n+m}$ obtained by permuting the first n letters and the last m letters in $[n+m]$ separately. If V is an ${\mathfrak {S}}_n$-module and W is an ${\mathfrak {S}}_m$-module, their induction product $V \circ W$ is the ${\mathfrak {S}}_{n+m}$-module given by

(2.11)$$ \begin{align} V \circ W = \mathrm{Ind}^{{\mathfrak{S}}_{n+m}}_{{\mathfrak{S}}_n \times {\mathfrak{S}}_m}(V \otimes W). \end{align} $$

Induction product and Frobenius image are related in that

(2.12)$$ \begin{align} {\mathrm {Frob}}(V \circ W) = {\mathrm {Frob}}(V) \cdot {\mathrm {Frob}}(W). \end{align} $$

Frobenius images interact with the skewing operators $h_j^{\perp }$ and $e_j^{\perp }$ in the following way. Let V be an ${\mathfrak {S}}_n$-module, let $1 \leq j \leq n$ and consider the parabolic subgroup ${\mathfrak {S}}_j \times {\mathfrak {S}}_{n-j}$ of ${\mathfrak {S}}_n$. We have the group algebra elements $\eta _j, \varepsilon _j \in {\mathbb {Q}}[{\mathfrak {S}}_j]$

(2.13)$$ \begin{align} \eta_j = \sum_{w \in {\mathfrak{S}}_j} w \quad \quad \quad \quad \varepsilon_j = \sum_{w \in {\mathfrak{S}}_j} {\mathrm {sign}}(w) \cdot w \end{align} $$

which symmetrise and antisymmetrise in the first j letters, respectively. Since $\eta _j$ and $\varepsilon _j$ commute with permutations in the second parabolic factor ${\mathfrak {S}}_{n-j}$, the vector spaces

(2.14)$$ \begin{align} \eta_j V = \{ \eta_j \cdot v \,:\, v \in V \} \quad \quad \quad \quad \varepsilon_j V = \{ \varepsilon_j \cdot v \,:\, v \in V \} \end{align} $$

are naturally ${\mathfrak {S}}_{n-j}$-modules. The Frobenius images of these modules are as follows.

The proof of Lemma 2.4, which we omit, uses Frobenius reciprocity. Lemma 2.4 may be generalised by considering the image of V under $\sum _{w \in {\mathfrak {S}}_j} \chi ^{\mu }(w) \cdot w \in {\mathbb {Q}}[{\mathfrak {S}}_j]$, where $\mu \vdash j$ is any partition and $\chi ^{\mu }: {\mathfrak {S}}_j \rightarrow {\mathbb {C}}$ is the irreducible character; the effect on Frobenius images is the operator $s_{\mu }^{\perp }$.

In this paper, we will consider (bi)graded vector spaces and modules. If $V = \bigoplus _{i \geq 0} V_i$ is a graded vector space with each piece $V_i$ finite-dimensional, recall that its Hilbert series is given by

(2.15)$$ \begin{align} {\mathrm {Hilb}}(V; q) = \sum_{i \geq 0} \dim (V_i) \cdot q^i. \end{align} $$

Similarly, if $V = \bigoplus _{i,j \geq 0} V_{i,j}$ is a bigraded vector space, we have the bigraded Hilbert series

(2.16)$$ \begin{align} {\mathrm {Hilb}}(V; q,z) = \sum_{i,j \geq 0} \dim(V_{i,j}) \cdot q^i z^j. \end{align} $$

If $V = \bigoplus _{i \geq 0} V_i$ is a graded ${\mathfrak {S}}_n$-module, its graded Frobenius image is

(2.17)$$ \begin{align} {\mathrm {grFrob}}(V; q) = \sum_{i \geq 0} {\mathrm {Frob}}(V_i) \cdot q^i. \end{align} $$

Extending this, if $V = \bigoplus _{i,j \geq 0} V_{i,j}$ is a bigraded ${\mathfrak {S}}_n$-module, its bigraded Frobenius image is

(2.18)$$ \begin{align} {\mathrm {grFrob}}(V; q,z) = \sum_{i,j \geq 0} {\mathrm {Frob}}(V_{i,j}) \cdot q^i z^j. \end{align} $$

2.4 Ordered set superpartitions

We will show that the duality modules ${\mathbb {W}}_{n,k}$ are governed by the combinatorics of ordered set superpartitions in ${\mathcal {OSP}}_{n,k}$. The more general modules ${\mathbb {W}}_{n,k,s}$ of Definition 2.1 which we will use to inductively describe the ${\mathbb {W}}_{n,k}$ are controlled by the following more general combinatorial objects.

Note that, when $s=k$, ${\mathcal {OSP}}_{n,k} = {\mathcal {OSP}}_{n,k,s}$. We refer to elements $\sigma = (B_1 \mid \cdots \mid B_k) \in {\mathcal {OSP}}_{n,k,s}$ as ordered set superpartitions, despite the fact that any of the last $k-s$ sets $B_{s+1}, \dots , B_k$ in $\sigma $ could be empty and that the minimal elements of $B_{s+1}, \dots , B_k$ (if they exist) may be barred.

3.1 The statistics ${\mathrm {coinv}}$ and ${\mathrm {codinv}}$

In this section, we define two statistics on ${\mathcal {OSP}}_{n,k,s}$. The first of these is an extension of the classical inversion statistic (or rather, its complement) on permutations in ${\mathfrak {S}}_n$. Given $\pi = \pi _1 \dots \pi _n \in {\mathfrak {S}}_n$, its coinversion code is the sequence $(c_1, \dots , c_n)$, where $c_i$ is the number of entries in the set $\{i+1, i+2, \dots , n \}$ which appear to the right of i in $\pi $. The coinversion number of $\pi $ is the sum ${\mathrm {coinv}}(\pi ) = c_1 + \cdots + c_n$. We generalise these concepts as follows.

Definition 3.1. Let $\sigma = (B_1 \mid \cdots \mid B_k) \in {\mathcal {OSP}}_{n,k,s}$ be an ordered set superpartition. The coinversion code is the length n sequence ${\mathrm {code}}(\sigma ) = (c_1, \dots , c_n)$ over the alphabet $\{0, 1, 2, \dots , \bar {0}, \bar {1}, \bar {2}, \dots \}$ whose $a^{th}$ entry $c_a$ is defined as follows. Suppose that a lies in the $i^{th}$ block $B_i$ of $\sigma $.

• • The entry $c_a$ is barred if and only if a is barred in $\sigma $.

• • If $a = \min B_i$ and $i \leq s$, then

  $$ \begin{align*} c_a = \left| \{ i+1 \leq j \leq s \,:\, \min B_j> a \} \right|. \end{align*} $$

• • If a is barred, then

  $$ \begin{align*} c_a = \begin{cases} | \{ 1 \leq j \leq i-1 \,:\, \min B_j < a \} | & \text{if } i \leq s, \\ | \{ 1 \leq j \leq s \,:\, \min B_j < a \} | + (i-s-1) & \text{if } i> s. \end{cases} \end{align*} $$

• • Otherwise, we set

  $$ \begin{align*} c_a = | \{ i+1 \leq j \leq s \,:\, \min B_j> a \} | + (i-1). \end{align*} $$

The coinversion number of $\sigma $ is the sum

$$ \begin{align*} {\mathrm {coinv}}(\sigma) = c_1 + \cdots + c_n \end{align*} $$

of the entries $(c_1, \dots , c_n)$ in ${\mathrm {code}}(\sigma )$.

For example, consider the ordered set superpartition

$$ \begin{align*} \sigma = ( 2, \, \bar{5} \mid 3, \, 6, \, \bar{8}, \, 9 \mid \varnothing \mid \bar{1}, \, \bar{4}, \, 7 \mid \varnothing) \in {\mathcal{OSP}}_{9,5,2}. \end{align*} $$

The coinversion code of $\sigma $ is given by

$$ \begin{align*} {\mathrm {code}}(\sigma) = (c_1, \dots, c_9) = ( \bar{1}, 1, 0, \bar{3}, \bar{0}, 1, 3, \bar{1}, 1) ,\end{align*} $$

so that

$$ \begin{align*} {\mathrm {coinv}}(\sigma) = 1 + 1 + 0 + 3 + 0 + 1 + 3 + 1 + 1 = 11. \end{align*} $$

Bars on the entries $c_1, \dots , c_n$ are ignored when calculating ${\mathrm {coinv}}(\sigma ) = c_1 + \cdots + c_n$.

The coinversion code $(c_1, \dots , c_n)$ can be visualised by considering the column diagram notation for ordered set superpartitions. Given $\sigma = (B_1 \mid \cdots \mid B_k) \in {\mathcal {OSP}}_{n,k,s}$, we draw the entries of $B_i$ in the $i^{th}$ column. The entries of $B_i$ fill the $i^{th}$ column according to the following rules.

• • For $1 \leq i \leq k$, the barred entries of $B_i$ start at height 1 and fill up in increasing order.

• • For $1 \leq i \leq s$, the unbarred entries of $B_i$ start at height 0 and fill down in increasing order.

• • For $s+1 \leq i \leq k$, the unbarred entries of $B_i$ start at height $-1$ and fill down in increasing order; we also place a $\bullet $ at height 0 in these columns.

In our example, the column diagram of

$$ \begin{align*} ( 2, \, \bar{5} \mid 3, \, 6, \, \bar{8}, \, 9 \mid \varnothing \mid \bar{1}, \, \bar{4}, \, 7 \mid \varnothing) \in {\mathcal{OSP}}_{9,5,2}. \end{align*} $$

is given by

where the column indices are shown below and the heights of entries are shown on the left. The three $\bullet $’s at the height 0 level correspond to the fact that the blocks $B_3, B_4, B_5$ are allowed to be empty for $\sigma \in {\mathcal {OSP}}_{9,5,2}$. Given $\sigma \in {\mathcal {OSP}}_{n,k,s}$, we have the following column diagram interpretation of the $a^{th}$ letter $c_a$ of ${\mathrm {code}}(\sigma ) = (c_1, \dots , c_n)$.

• • If a appears at height 0, then $c_a$ counts the number of height zero entries to the right of a which are $> a$.

• • If a appears at negative height in column i, then $c_a$ is $i-1$, plus the number of height 0 entries to the right of a which are $> a$.

• • If a appears at positive height, then $c_a$ counts the number of height 0 entries to the left of a which are $< a$, plus the number of $\bullet $’s to the left of a.

We will see that codes $(c_1, \dots , c_n)$ of elements $\sigma \in {\mathcal {OSP}}_{n,k,s}$ correspond to a monomial basis of the quotient ring ${\mathbb {W}}_{n,k,s}$. In order to obtain the bigraded Frobenius image of ${\mathbb {W}}_{n,k,s}$ in terms of ${\mathrm {coinv}}$, we use diagrams to define a formal power series as follows.

The reading word of $\sigma \in {\mathcal {OSP}}_{n,k,s}$, denoted ${\mathrm {read}}(\sigma )$ is the permutation in ${\mathfrak {S}}_n$ obtained by reading the column diagram of $\sigma $ from top to bottom, and, within each row, from right to left (ignoring any bars on letters). If $\sigma $ is our example ordered set partition above, we have

$$ \begin{align*} {\mathrm {read}}(\sigma) = 418532769 \in {\mathfrak{S}}_9. \end{align*} $$

Definition 3.2. Let $n, k, s \geq 0$ be integers, and let $0 \leq r \leq n-s$. We define a quasisymmetric function $C^{(r)}_{n,k,s}({\mathbf {x}};q)$ by the formula

(3.1)$$ \begin{align} C^{(r)}_{n,k,s}({\mathbf {x}};q) = \sum_{\sigma \in {\mathcal{OSP}}_{n,k,s}^{(r)}} q^{{\mathrm {coinv}}(\sigma)} \cdot F_{{\mathrm {iDes}}({\mathrm {read}}(\sigma)),n}({\mathbf {x}}). \end{align} $$

Also define a quasisymmetric function $C_{n,k,s}({\mathbf {x}};q,z)$ by

(3.2)$$ \begin{align} C_{n,k,s}({\mathbf {x}};q,z) = \sum_{r = 0}^{n-s} C^{(r)}_{n,k,s}({\mathbf {x}};q) \cdot z^r. \end{align} $$

We may avoid the use of the F’s in the definition of the C-functions by allowing repeated entries in our column diagrams. Let ${\mathcal {CT}}_{n,k,s}$ be the family of column tableaux which are all fillings of finite subsets of the infinite strip $[k] \times \infty $ with the alphabet $\{ \bullet , 1, 2, \dots , \bar {1}, \bar {2}, \dots \}$, such that

• • The height 0 row is filled with a sequence of s unbarred numbers, followed by a sequence of $k-s \bullet $’s.

• • The filled cells form a contiguous sequence within each column.

• • Entries at positive height are barred, while entries at negative height are unbarred.

• • Unbarred numbers weakly increase going down, and barred numbers strictly increase going up.

• • An unbarred number at height 0 is strictly smaller than any barred number above it.

An example column tableau $\tau \in {\mathcal {CT}}_{13, 5, 2}$ is shown below.

The coinversion number ${\mathrm {coinv}}(\tau )$ extends naturally to column tableaux $\tau \in {\mathcal {CT}}_{n,k,s}$: given an entry a in such a tableau, we may compute its contribution $c_a$ to ${\mathrm {coinv}}$ as before, and sum over all entries. For example, the column tableau depicted above has coinversion number

$$ \begin{align*}3 + 1 + 0 + 1 + 1 + 0 + 0 + 1 + 4 + 3 + 1 + 4 + 1 = 20,\end{align*} $$

where the entries in the sum on the left are processed in reading order, that is the first value in the sum is 3 because the $\bar {4}$ in column 4 contributes 3 to the coinversion number. Let ${\mathbf {x}}^{\tau } = x_1^{\alpha _1} x_2^{\alpha _2} \cdots $, where $\alpha _i$ is the number of $i's$ in $\tau $. In our example, we have ${\mathbf {x}}^{\tau } = x_1^3 x_2 x_3^3 x_4^3 x_5 x_6 x_7$. The exponent sequence $\alpha = (\alpha _1, \alpha _2, \dots )$ of ${\mathbf {x}}^{\tau }$ is called the content of $\tau $. Let ${\mathcal {CT}}_{n,k,s}^{(r)} \subseteq {\mathcal {CT}}_{n,k,s}$ be the subfamily of column tableaux with r barred letters.

Observation 3.3. The formal power series $C_{n,k,s}^{(r)}({\mathbf {x}};q)$ and $C_{n,k,s}({\mathbf {x}};q,z)$ are given by

$$ \begin{align*} C_{n,k,s}^{(r)}({\mathbf {x}};q) = \sum_{\tau \in {\mathcal {CT}}_{n,k,s}^{(r)}} q^{{\mathrm {coinv}}(\tau)} {\mathbf {x}}^{\tau} \quad \text{and} \quad C_{n,k,s}({\mathbf {x}};q,z) =\sum_{\tau \in {\mathcal {CT}}_{n,k,s}} q^{{\mathrm {coinv}}(\tau)} z^{\# \text{ of bars in } \tau} {\mathbf {x}}^{\tau}. \end{align*} $$

This observation follows from the definition of the fundamental quasisymmetric functions. The formulas for the C-functions in Observation 3.3 are more aesthetic but less efficient than those in Definition 3.2. It will turn out that $C_{n,k,s}({\mathbf {x}};q,z)$ is the bigraded Frobenius image ${\mathrm {grFrob}}({\mathbb {W}}_{n,k,s};q,z)$. For reasons related to skewing recursions, the refinement $C^{(r)}_{n,k,s}({\mathbf {x}};q)$ will be convenient to consider. At this point, it is not clear that the C-functions are even symmetric. Their symmetry (and Schur positivity) will follow from an alternative description in terms of another statistic on ${\mathcal {OSP}}_{n,k,s}$.

Let $\sigma \in {\mathcal {OSP}}_{n,k,s}$. We augment the column diagram of $\sigma $ by placing infinitely many $+ \infty $’s below the entries in every column (we drop the $+$ in diagrams for the sake of compactness). Furthermore, we regard every $\bullet $ in the column diagram as being filled with a 0. Ignoring bars, a pair of entries $a < b$ form a diagonal coinversion pair if

• • a appears to the left of and at the same height as b, or

• • a appears to the right of and at height one less than b.

Schematically, these conditions have the form

for $a < b$.

When $n=k=s$, ${\mathrm {codinv}}$ and ${\mathrm {coinv}}$ are identical. Although these statistics clearly differ in general, it will turn out that ${\mathrm {codinv}}$ is equidistributed with ${\mathrm {coinv}}$ on ${\mathcal {OSP}}_{n,k,s}$ and even on the subsets ${\mathcal {OSP}}_{n,k,s}^{(r)}$ obtained by restricting to r barred letters. In analogy with the C-functions, we define a quasisymmetric function attached to ${\mathrm {codinv}}$.

Definition 3.5. Let $n, k, s \geq 0$ be integers, and let $0 \leq r \leq n-s$. Define a quasisymmetric function $D^{(r)}_{n,k,s}({\mathbf {x}};q)$ by the formula

(3.3)$$ \begin{align} D^{(r)}_{n,k,s}({\mathbf {x}};q) = \sum_{\sigma \in {\mathcal{OSP}}_{n,k,s}^{(r)}} q^{{\mathrm {codinv}}(\sigma)} \cdot F_{{\mathrm {iDes}}({\mathrm {read}}(\sigma)),n}({\mathbf {x}}). \end{align} $$

Also define

(3.4)$$ \begin{align} D_{n,k,s}({\mathbf {x}};q,z) = \sum_{r = 0}^{n-s} D^{(r)}_{n,k,s}({\mathbf {x}};q) \cdot z^r. \end{align} $$

Like the C-functions, the D-functions may be expressed in terms of infinite sums over column tableaux.

Observation 3.6. The formal power series $D_{n,k,s}^{(r)}({\mathbf {x}};q)$ and $D_{n,k,s}({\mathbf {x}};q,z)$ are given by

$$ \begin{align*} D_{n,k,s}^{(r)}({\mathbf {x}};q) = \sum_{\tau \in {\mathcal {CT}}_{n,k,s}^{(r)}} q^{{\mathrm {codinv}}(\tau)} {\mathbf {x}}^{\tau} \quad \text{and} \quad D_{n,k,s}({\mathbf {x}};q,z) = \sum_{\tau \in {\mathcal {CT}}_{n,k,s}} q^{{\mathrm {codinv}}(\tau)} z^{\# \text{ of bars in } \tau} {\mathbf {x}}^{\tau}. \end{align*} $$

We use the theory of LLT polynomials to show that the D-functions are symmetric and Schur positive.

Proof. We prove this fact by writing $D_{n,k,s}({\mathbf {x}}; q, z)$ as a positive linear combination of LLT polynomials [Reference Lascoux, Leclerc and Thibon18]. We use the version of LLT polynomials employed by Haglund, Haiman and Loehr [Reference Haglund, Haiman and Loehr11].

A skew diagram is a set of cells in the first quadrant given by $\mu / \nu $ for some partitions $\mu \supseteq \nu $. Given a tuple $ {\boldsymbol\lambda } = (\lambda ^{(1)}, \lambda ^{(2)}, \ldots , \lambda ^{(l)})$ of skew diagrams, the LLT polynomial indexed by ${\boldsymbol\lambda }$ is

$$ \begin{align*} \mathrm{LLT}_{\boldsymbol\lambda}({\mathbf {x}}; q) = \sum q^{\mathrm{inv}(T)} {\mathbf {x}}^T, \end{align*} $$

where the sum is over all semistandard fillings of the skew diagrams of $ {\boldsymbol\lambda }$, ${\mathbf {x}}^T$ is the product, where $x_i$ appears as many times as i appears in the filling T, and ${\mathrm {inv}}(T)$ is the following statistic. Given a cell u with coordinates $(x, y)$ in one of the skew shapes $\lambda ^{(i)}$, where the leftmost cell at height 0 has coordinates (0,0), the content of u is $c(u) = x - y$. Then ${\mathrm {inv}}(T)$ is the number of pairs of cells $u \in \lambda ^{(i)}$, $v \in \lambda ^{(j)}$ with $i < j$, such that

• • $c(u) = c(v)$ and $T(u)> T(v)$, or

• • $c(u) + 1 = c(v)$ and $T(u) < T(v)$.

LLT polynomials are Schur-positive symmetric functions [Reference Grojnowski and Haiman10, Reference Haglund, Haiman and Loehr11, Reference Lascoux, Leclerc and Thibon18]. We claim that we can decompose $D_{n,k,s}^{(r)}({\mathbf {x}};q)$ into LLT polynomials:

(3.5)$$ \begin{align} D_{n,k,s}^{(r)}({\mathbf {x}}; q) = \sum_{\boldsymbol\lambda} q^{\mathrm{stat}({\boldsymbol\lambda})} \mathrm{LLT}_{{\boldsymbol\lambda}}({\mathbf {x}}; q), \end{align} $$

where $\mathrm {stat}$ is a fixed statistic depending on $ {\boldsymbol\lambda }$ and the sum is over all tuples of skew diagrams $ {\boldsymbol\lambda } = (\lambda ^{(1)}, \ldots , \lambda ^{(k)})$ satisfying

• • if $i \leq k-s$, then $\lambda ^{(i)} = \mu ^{(i)} / (1)$ for a hook shape $\mu ^{(i)}$,

• • if $i> k-s$, then $\lambda ^{(i)}$ is a single nonempty hook shape,

• • $\sum _{i=1}^{k} |\lambda ^{(i)}| = n$ and

• • $ {\boldsymbol\lambda }$ has r total cells of negative content.

We define nonnegative integer sequences $\alpha , \beta \in \mathbb {N}^{k}$ so that

• • $\lambda ^{(i)} = (\alpha _i + 1, 1^{\beta _i}) / (1)$ for $i \leq k-s$ and

• • $\lambda ^{(i)} = (\alpha _i + 1, 1^{\beta _i})$ for $i> k-s$.

Given a specific semistandard filling T that contributes to $\mathrm {LLT}_{{\boldsymbol\lambda }}({\mathbf {x}}; q)$ for such a $ {\boldsymbol\lambda }$, we will create a column tableau $\tau \in {\mathcal {CT}}_{n,k,s}^{(r)}$, such that $\tau $ will have exactly $\alpha _i$ entries at negative height and $\beta _i$ entries at positive height in column $k-i+1$. The sequences $\alpha $ and $\beta $ will completely determine how many diagonal coinversions involving an $\infty $ or a $\bullet $, respectively, appear in $\tau $. In Figure 1, there are six such diagonal coinversions involving an $\infty $ and seven involving a $\bullet $. In general, this number is

$$ \begin{align*} \mathrm{stat}({\boldsymbol\lambda}) = \sum_{1 \leq i < j \leq k}\left( |\alpha_i - \alpha_j| - \chi(\alpha_i> \alpha_j) \right) + \sum_{i =1}^{k-s} | \{j > i: \beta_j \neq 0 \}| , \end{align*} $$

where $\chi (p)$ is 1 if p is true and $0$ if p is false. Note that $\mathrm {stat}( {\boldsymbol\lambda })$ can be computed from $\alpha $ and $\beta $ only.

Figure 1 We depict an example of the correspondence in the proof of Proposition 3.7. Each of the five skew diagrams on the left is justified so that its bottom left entry has content 0. In this example, we have $\alpha = (2, 1, 0, 2, 0)$, $\beta = (0, 2, 0, 2, 2)$ and $\mathrm {stat}( {\boldsymbol\lambda }) = 6 + 7 = 13$

Now, given a specific semistandard filling T of $ {\boldsymbol\lambda }$, we map the unique entry in $\lambda ^{(i)}$ with content j to column $k-i+1$ and row $-j$ in $\tau $. Every pair which contributes to ${\mathrm {inv}}(T)$ now contributes a diagonal coinversion between integers (not $\bullet $ or $\infty $) in $\tau $. In Figure 1, there are 10 such diagonal coinversions and the filling T has ${\mathrm {inv}}(T) = 10$. Since this map is bijective, we have the desired result.

3.2 Equality of the C- and D-functions

Our first main result states that the C-functions and D-functions coincide. This given, Proposition 3.7 implies that the C-functions are symmetric and Schur positive. We know of no direct proof of either of these facts.

Theorem 3.8. For any integers $n,k,s \geq 0$ with $k \geq s$ and any $0 \leq r \leq n-s$, we have the equality of formal power series

(3.6)$$ \begin{align} C_{n,k,s}^{(r)}({\mathbf {x}};q) = D_{n,k,s}^{(r)}({\mathbf {x}};q). \end{align} $$

Consequently, we have

(3.7)$$ \begin{align} C_{n,k,s}({\mathbf {x}};q,z) = D_{n,k,s}({\mathbf {x}};q,z). \end{align} $$

Our proof of Theorem 3.8 is combinatorial and uses the column tableau forms of the formal power series $C_{n,k,s}^{(r)}({\mathbf {x}};q)$ and $D_{n,k,s}^{(r)}({\mathbf {x}};q)$ in Observations 3.3 and 3.6. The idea is to consider building up a general column tableau $\tau \in {\mathcal {CT}}_{n,k,s}^{(r)}$ from the column tableau consisting of $k-s$ empty columns by successively adding larger entries and showing that the statistics ${\mathrm {coinv}}$ and ${\mathrm {codinv}}$ satisfy the same recursions.

Proof. Let $\tau $ be a column tableau with k columns and $k-s \bullet $’s whose entries are $< N$. We consider building a larger column tableau involving N’s from $\tau $ by the following three-step process.

1. 1. Placing $\bar {N}$’s on top of some subset of the k columns of $\tau $.

2. 2. Placing N’s at height 0 between and on either side of the s columns of $\tau $ without a $\bullet $.

3. 3. Placing N’s at negative heights below some multiset of columns of the resulting figure.

We track the behavior of ${\mathrm {coinv}}$ and ${\mathrm {codinv}}$ as we perform this procedure, starting by placing the $\bar {N}$’s on top of columns.

Placing a $\bar {N}$ on top of column i of $\tau $ increases the statistic ${\mathrm {coinv}}$ by $i-1$. We reflect this fact by giving column i the barred ${\mathrm {coinv}}$ label of $i-1$. The barred ${\mathrm {coinv}}$ labels of the column tableau are shown in bold and barred below.

The barred ${\mathrm {coinv}}$ labels give rise to a bijection

(3.8)$$ \begin{align} \iota^{{\mathrm {coinv}}}_{\bar{N}}: \left\{ \begin{array}{c}k\text{-column tableaux } \tau \text{ with} \\ k-s \bullet\text{'s and all entries } < N \end{array} \right\} &\times \left\{\begin{array}{c}\text{subsets } S \text{ of} \\ \{0,1,\dots,k-1\}\end{array} \right\} \longrightarrow \notag \\ &\qquad\qquad\quad\left\{ \begin{array}{c}k\text{-column tableaux } \tau' \text{ with } k-s \bullet\text{'s,} \\ \text{such that all entries of } \tau' \text{ are } \leq N \text{ and}\\ \text{the only } N\text{'s in } \tau' \text{ are barred}\end{array} \right\} \end{align} $$

by letting $\iota ^{{\mathrm {coinv}}}_{\bar {N}}(\tau ,S)$ be the tableau $\tau '$ obtained by placing a $\bar {N}$ on top of every column with barred ${\mathrm {coinv}}$ label in S. If $\iota ^{{\mathrm {coinv}}}_{\bar {N}}: (\tau ,S) \mapsto \tau '$, then ${\mathrm {coinv}}(\tau ') = {\mathrm {coinv}}(\tau ) + \sum _{i \in S} i$.

Next, we consider the effect of $\bar {N}$ insertion on ${\mathrm {codinv}}$. We bijectively label the k columns of $\tau $ with the k barred ${\mathrm {codinv}}$ labels $0, 1, \dots , k-1$ (in that order) in descending order of maximal height and, within columns of the same maximal height, proceeding from right to left. The barred ${\mathrm {codinv}}$ labels in our example are shown below.

The barred ${\mathrm {codinv}}$ labels give a bijection

(3.9)$$ \begin{align} \iota^{{\mathrm {codinv}}}_{\bar{N}}: \left\{ \begin{array}{c}k\text{-column tableaux } \tau \text{ with} \\ k-s \bullet\text{'s and all entries } < N \end{array} \right\} &\times \left\{ \begin{array}{c}\text{subsets } S \text{ of} \\ \{0,1,\dots,k-1\}\end{array} \right\} \longrightarrow \notag \\ &\qquad\qquad\quad \left\{ \begin{array}{c}k\text{-column tableaux } \tau' \text{ with } k-s \bullet\text{'s,} \\ \text{such that all entries of } \tau' \text{ are } \leq N \text{ and} \\ \text{the only } N\text{'s in } \tau' \text{ are barred}\end{array} \right\} , \end{align} $$

where $\iota ^{{\mathrm {codinv}}}_{\bar {N}}(\tau ,S)$ is obtained from $\tau $ by placing a $\bar {N}$ on top of every column with barred ${\mathrm {codinv}}$ label indexed by S. If $\iota ^{{\mathrm {codinv}}}_{\bar {N}}: (\tau ,S) \mapsto \tau '$, then ${\mathrm {codinv}}(\tau ') = {\mathrm {codinv}}(\tau ) + \sum _{i \in S} i$.

We move on to Step 2 of our insertion procedure: creating new columns by placing N’s at height 0. A single ‘height 0 insertion map’ $\iota _0$ of this kind on column tableaux has the same effect on ${\mathrm {coinv}}$ and ${\mathrm {codinv}}$.

If $\tau $ is a column tableau with s nonbullet letters at height zero, N’s can be placed (with repetition) in any of the $s+1$ places between and on either side of these nonbullet letters. This gives rise to a bijection

(3.10)$$ \begin{align} \iota_0: \left\{ \begin{array}{c}k\text{-column tableaux } \tau' \text{ with } k-s \bullet\text{'s,} \\ \text{such that all entries of } \tau' \text{ are } \leq N \text{ and}\\ \text{the only } N\text{'s in } \tau' \text{ are barred}\end{array} \right\} &\times \left\{ \begin{array}{c} \text{finite multisets } S \\ \text{drawn from } \{0,1,\dots,s\} \end{array} \right\} \longrightarrow \notag \\ &\bigsqcup_{K \geq k} \left\{ \begin{array}{c}K\text{-column tableaux } \tau" \text{ with } k-s \bullet\text{'s,} \\ \text{such that all entries of } \tau" \text{ are } \leq N \text{ and}\\ \text{no } N\text{'s in } \tau" \text{ have negative height}\end{array} \right\} , \end{align} $$

where $\iota _0(\tau ',S)$ is obtained from $\tau '$ by inserting $m_i$ copies of N after column i of $\tau $, where $m_i$ is the multiplicity of i in S. Suppose $\iota _0(\tau ',S) = \tau "$. The number K of columns of $\tau "$ is related to the number k of columns of $\tau '$ by $K = k + |S|$. Furthermore, if there are b unbarred entries in $\tau '$ of negative height, we have

(3.11)$$ \begin{align} {\mathrm {coinv}}(\tau") = {\mathrm {coinv}}(\tau') + b \cdot |S| + \sum_{i \in S} i \quad \text{and} \quad {\mathrm {codinv}}(\tau") = {\mathrm {codinv}}(\tau') + b \cdot |S| + \sum_{i \in S} i. \end{align} $$

In other words, the height zero insertion map $\iota _0$ has the same effect on ${\mathrm {coinv}}$ and ${\mathrm {codinv}}$.

Finally, let $\tau "$ be a column tableau with K columns with entries $\leq N$ in which there are no N’s with negative height. To perform Step 3 of our insertion process, we insert N’s at the bottom of some multiset of columns of $\tau "$. We track the effect on ${\mathrm {coinv}}$ and ${\mathrm {codinv}}$ as before.

We label the columns of $\tau "$ from left-to-right with the unbarred ${\mathrm {coinv}}$ labels $0,1, \dots , K$. An example of this labeling is shown below in bold. Since we add unbarred letters on the bottom of a tableau, we show the unbarred labels there as well.

The unbarred ${\mathrm {coinv}}$ labels give a bijection

(3.12)$$ \begin{align} \iota^{{\mathrm {coinv}}}_N: \left\{ \begin{array}{c}K\text{-column tableaux } \tau" \text{ with } k-s \bullet\text{',} \\ \text{such that all entries of } \tau" \text{ are } \leq N \text{ and}\\ \text{no } N\text{'s in } \tau" \text{ have negative height}\end{array} \right\} &\times \left\{ \begin{array}{c} \text{finite multisets } S \\ \text{drawn from } \{0,1,\dots,K-1\} \end{array} \right\} \longrightarrow \notag \\ &\left\{ \begin{array}{c} K\text{-column tableaux } \tau"' \text{ with } k-s \bullet\text{'s,} \\ \text{such that all entries in } \tau"' \text{ are } \leq N \end{array} \right\} , \end{align} $$

where $\iota ^{{\mathrm {coinv}}}_N(\tau ",S)$ is obtained by placing $m_i$ copies of N below the column with label i, where $m_i$ is the multiplicity of i in S. If $\iota ^{{\mathrm {coinv}}}_N: (\tau ",S) \mapsto \tau "'$, then ${\mathrm {coinv}}(\tau "') = {\mathrm {coinv}}(\tau ") + \sum _{i \in S} i$.

The effect of inserting N’s at negative height on ${\mathrm {codinv}}$ may be described as follows. We label the columns of $\tau $ bijectively with the unbarred ${\mathrm {codinv}}$ labels $0,1,\dots ,k-1$ (in that order) starting at columns of lesser maximal depth and, within columns of the same maximal depth, proceeding from right to left. The unbarred ${\mathrm {codinv}}$ labels in our example as shown below.

The unbarred ${\mathrm {codinv}}$ labels give a bijection

(3.13)$$ \begin{align} \iota^{{\mathrm {codinv}}}_N: \left\{ \begin{array}{c}K\text{-column tableaux } \tau" \text{ with } k-s \bullet\text{'s,} \\ \text{such that all entries of } \tau" \text{ are } \leq N \text{ and}\\ \text{no } N\text{'s in } \tau" \text{ have negative height}\end{array} \right\} &\times \left\{\begin{array}{c} \text{finite multisets } S \\ \text{drawn from } \{0,1,\dots,K-1\} \end{array} \right\} \longrightarrow \notag\\ &\left\{ \begin{array}{c} K\text{-column tableaux } \tau"' \text{ with } k-s \bullet\text{'s,} \\ \text{such that all entries in } \tau"' \text{ are } \leq N \end{array} \right\} \end{align} $$

as follows. Given $(\tau ",S)$, we process the entries of S in weakly increasing order and, given an entry i, we place an N at the bottom of the column with unbarred ${\mathrm {codinv}}$ label i. Recalling that we consider every unfilled cell below a column to have the label $+ \infty $ for the purpose of calculating ${\mathrm {codinv}}$, we see that if $\iota ^{{\mathrm {codinv}}}_N: (\tau ",S) \mapsto \tau "'$, then ${\mathrm {codinv}}(\tau "') = {\mathrm {codinv}}(\tau ") + \sum _{i \in S} i$. In summary, the $\iota $-maps show that the C-functions and D-functions satisfy the same recursion, which establishes the theorem.

Although the D-functions are more directly seen to be symmetric and Schur positive, in order to describe a recursive formula for the action of $h_j^{\perp }$, it will be more convenient to use the C-functions instead (Lemma 3.14). This formula will involve the more refined $C_{n,k,s}^{(r)}({\mathbf {x}};q)$ rather than their coarsened versions $C_{n,k,s}({\mathbf {x}};q,z)$.

3.3 Substaircase shuffles

Given $\sigma \in {\mathcal {OSP}}_{n,k,s}$, ${\mathrm {code}}(\sigma ) = (c_1, \dots , c_n)$ is a length n sequence over the alphabet $\{0,1,2,\dots ,\bar {0},\bar {1},\bar {2},\dots \}$. In this subsection, we show that $\sigma \mapsto {\mathrm {code}}(\sigma )$ is injective and characterise its image. We begin by defining a partial order on sequences of (potentially barred) nonnegative integers.

The image of the classical coinversion code map on permutations in ${\mathfrak {S}}_n$ is given by words $(c_1, \dots , c_n)$ which are $\leq $ the ‘staircase’ word $(n-1, n-2, \dots , 1, 0)$. For ordered set partitions of $[n]$ into k blocks with no barred letters, this result was generalised in [Reference Rhoades and Wilson26], where the appropriate notion of ‘staircase’ is given by a shuffle of two sequences. We extend these definitions to barred letters as follows.

Recall that a shuffle of two sequences $(a_1, \dots , a_r)$ and $(b_1, \dots , b_s)$ is an interleaving $(c_1, \dots , c_{r+s})$ of these sequences which preserves the relative order of the a’s and the b’s. The following collection ${\mathcal {SS}}^{(r)}_{n,k,s}$ of words will turn out to be the image of the map ${\mathrm {code}}$ on ${\mathcal {OSP}}_{n,k,s}^{(r)}$.

Definition 3.10. Let $n, k, s \geq 0$ be integers. For $0 \leq r \leq n-s$, a staircase with r barred letters is a length n word obtained by shuffling

$$ \begin{align*} ( \overline{k-s-1} )^{r_0} (s-1) ( \overline{k-s} )^{r_1} (s-2) ( \overline{k-s+1} )^{r_2} (s-3) \cdots 0 ( \overline{k-1})^{r_s} \quad \text{and} \quad (k-1)^{n-r-s},\\[-16pt] \end{align*} $$

where $r_0 + r_1 + \cdots + r_s = r$. If $k=s$, we insist that $r_0 = 0$ in the above expression. Let ${\mathcal {SS}}^{(r)}_{n,k,s}$ be the family of words $(c_1, \dots , c_n)$ which are $\leq $ some staircase $(b_1, \dots , b_n)$. We also let

(3.14)$$ \begin{align} {\mathcal{SS}}_{n,k,s} = {\mathcal{SS}}_{n,k,s}^{(0)} \sqcup {\mathcal{SS}}_{n,k,s}^{(1)} \sqcup \cdots \sqcup {\mathcal{SS}}_{n,k,s}^{(n-s)}\\[-16pt]\nonumber \end{align} $$

and refer to words in ${\mathcal {SS}}_{n,k,s}$ as substaircase.

Definition 3.10 implies that

(3.15)$$ \begin{align} {\mathcal{SS}}_{n,k,s} = \varnothing \quad \quad \text{if } n < s.\\[-16pt]\nonumber \end{align} $$

We give an example to clarify these concepts.

Example 3.11. Consider the case $n = 5, k = 3, s = 2$ and $r = 2$. The staircases with r barred letters are the shuffles of any of the six words

$$ \begin{align*} ( \bar{0}, \bar{0}, 1, 0), \, \, \, ( \bar{0}, 1, \bar{1}, 0), \, \, \, (\bar{0}, 1, 0, \bar{2}), \, \, \, (1, \bar{1}, \bar{1}, 0), \, \, \, (1, \bar{1}, 0, \bar{2}), \, \, \, (1, 0, \bar{2}, \bar{2})\\[-16pt] \end{align*} $$

with the single-letter word $(2)$. For example, if we shuffle $(2)$ into the second sequence from the left, we get the five staircases

$$ \begin{align*} (2, \bar{0}, 1, \bar{1}, 0), \, \, \, (\bar{0}, 2, 1, \bar{1}, 0), \, \, \, (\bar{0}, 1, 2, \bar{1}, 0), \, \, \, (\bar{0}, 1, \bar{1}, 2, 0), \, \, \, (\bar{0}, 1, \bar{1}, 0, 2). \end{align*} $$

The leftmost sequence displayed above contributes

$$ \begin{align*} (2, \bar{0}, 1, \bar{1}, 0), \, \, \, (1, \bar{0}, 1, \bar{1}, 0), \, \, \, (0, \bar{0}, 1, \bar{1}, 0), \, \, \, (2, \bar{0}, 0, \bar{1}, 0), \, \, \, (1, \bar{0}, 0, \bar{1}, 0), \, \, \, (0, \bar{0}, 0, \bar{1}, 0), \, \, \, \end{align*} $$

$$ \begin{align*} (2, \bar{0}, 1, \bar{0}, 0), \, \, \, (1, \bar{0}, 1, \bar{0}, 0), \, \, \, (0, \bar{0}, 1, \bar{0}, 0), \, \, \, (2, \bar{0}, 0, \bar{0}, 0), \, \, \, (1, \bar{0}, 0, \bar{0}, 0), \, \, \, (0, \bar{0}, 0, \bar{0}, 0) \, \, \, \end{align*} $$

to ${\mathcal {SS}}_{5,3,2}^{(2)}$. These are the 12 sequences which have the same bar pattern as, and are componentwise $\leq $ to, the staircase $(2, \bar {0}, 1, \bar {1}, 0)$.

The previous example shows that applying Definition 3.10 to obtain the set of words in ${\mathcal {SS}}_{n,k,s}$ can be involved. The following lemma gives a simple recursive definition of substaircase sequences of length n in terms of substaircase sequences of length $n-1$. This recursion gives an efficient way to calculate ${\mathcal {SS}}_{n,k,s}$ and will be useful in our algebraic analysis of ${\mathbb {W}}_{n,k,s}$ in Section 4. It will turn out that words in ${\mathcal {SS}}_{n,k,s}$ index a monomial basis of ${\mathbb {W}}_{n,k,s}$.

Lemma 3.12. Let $n, k, s \geq 0$ be integers with $k \geq s$, and let $0 \leq r \leq n-s$. The set ${\mathcal {SS}}_{n,k,s}^{(r)}$ has the disjoint union decomposition

(3.16)$$ \begin{align} {\mathcal{SS}}_{n,k,s}^{(r)} = &\bigsqcup_{a = 0}^{k-s-1} \left\{ ( \bar{a}, c_2, \dots, c_n) \,:\, (c_2, \dots, c_n) \in {\mathcal{SS}}_{n-1,k,s}^{(r-1)} \right\} \, \sqcup \end{align} $$

(3.17)$$ \begin{align} & \bigsqcup_{a = 0}^{s-1} \left\{ (a, c_2, \dots, c_n) \,:\, (c_2, \dots, c_n) \in {\mathcal{SS}}_{n-1,k,s-1}^{(r)} \right\} \, \sqcup \end{align} $$

(3.18)$$ \begin{align} & \bigsqcup_{a = s}^{k-1} \left\{ (a, c_2, \dots, c_n) \,:\, (c_2, \dots, c_n) \in {\mathcal{SS}}_{n-1,k,s}^{(r)} \right\} ,\end{align} $$

and the set ${\mathcal {SS}}_{n,k,s}$ has the disjoint union decomposition

(3.19)$$ \begin{align} {\mathcal{SS}}_{n,k,s} = &\bigsqcup_{a = 0}^{k-s-1} \left\{ ( \bar{a}, c_2, \dots, c_n) \,:\, (c_2, \dots, c_n) \in {\mathcal{SS}}_{n-1,k,s} \right\} \, \sqcup \end{align} $$

(3.20)$$ \begin{align} & \bigsqcup_{a = 0}^{s-1} \left\{ (a, c_2, \dots, c_n) \,:\, (c_2, \dots, c_n) \in {\mathcal{SS}}_{n-1,k,s-1} \right\} \, \sqcup \end{align} $$

(3.21)$$ \begin{align} & \bigsqcup_{a = s}^{k-1} \left\{ (a, c_2, \dots, c_n) \,:\, (c_2, \dots, c_n) \in {\mathcal{SS}}_{n-1,k,s} \right\}. \end{align} $$

Proof. The second disjoint union decomposition for ${\mathcal {SS}}_{n,k,s}$ follows from the first disjoint union decomposition for ${\mathcal {SS}}_{n,k,s}^{(r)}$ by taking the (disjoint) union over all r, so we focus on the first decomposition.

Given $(c_1, c_2, \dots , c_n) \in {\mathcal {SS}}_{n,k,s}^{(r)}$, there exists some word $(b_1, b_2, \dots , b_n)$ obtained by shuffling

$$ \begin{align*} ( \overline{k-s-1} )^{r_0} (s-1) ( \overline{k-s} )^{r_1} (s-2) ( \overline{k-s+1} )^{r_2} (s-3) \cdots 0 ( \overline{k-1})^{r_s} , \end{align*} $$

where $r_0 + r_1 + \cdots + r_s = r$ with the constant sequence $(k-1)^{n-r-s}$, such that $(c_1, c_2, \dots , c_n)$ has the same bar pattern as $(b_1, b_2, \dots , b_n)$ and $c_i \leq b_i$ for all i. There are three possibilities for the first entry $c_1$.

• • If $c_1 = \bar {a}$ is barred, then we must have $r_0> 0$ and $b_1 = \overline {k-s-1}$. It follows that $0 \leq a \leq k-s-1$. Furthermore, the word $(b_2, \dots , b_n)$ is a shuffle of

  $$ \begin{align*} ( \overline{k-s-1} )^{r_0-1} (s-1) ( \overline{k-s} )^{r_1} (s-2) ( \overline{k-s+1} )^{r_2} (s-3) \cdots 0 ( \overline{k-1} )^{r_s} \quad \text{and} \quad (k-1)^{n-r-s}. \end{align*} $$
  This implies that $(c_2, \dots , c_n) \in {\mathcal {SS}}_{n-1,k,s}^{(r-1)}$. Conversely, given $(c_2, \dots , c_n) \in {\mathcal {SS}}_{n-1,k,s}^{(r-1)}$ and $0 \leq a \leq k-s-1$, we see that $(\bar {a}, c_2, \dots , c_n) \in {\mathcal {SS}}_{n,k,s}^{(r)}$ by prepending the letter $\overline {k-s-1}$ to any $(n-1,k,s)$-staircase $(b_2, \dots , b_n) \geq (c_2, \dots , c_n)$.

• • If $c_1 = a$ is unbarred with $0 \leq a \leq s - 1$, the first letter $b_1$ of $(b_1, b_2, \dots , b_n)$ must also be unbarred and $r_0 = 0$. This means that $b_1 = s-1$ or $k-1$. If $k> s$ and $b_1 = k-1$, we may interchange the $b_1$ with the unique occurrence of $s-1$ in $(b_1, b_2, \dots , b_n)$ to get a new staircase $(b^{\prime }_1, b^{\prime }_2, \dots , b^{\prime }_n)$ which shares the color pattern of $(c_1, c_2, \dots , c_n)$ and satisfies $c_i \leq b^{\prime }_i$ for all i. We may, therefore, assume that $b_1 = s-1$. This means that the sequence $(b_2, \dots , b_n)$ is a shuffle of the words

  $$ \begin{align*} ( \overline{k-s} )^{r_1} (s-2) ( \overline{k-s+1} )^{r_2} (s-3) \cdots 0 ( \overline{k-1} )^{r_s} \quad \text{and} \quad (k-1)^{n-r-s} \end{align*} $$
  and $r_1 + r_2 + \cdots + r_s = r$. Therefore, we have $(c_2, \dots , c_n) \in {\mathcal {SS}}_{n-1,k,s-1}^{(r)}$. Conversely, given $(c_2, \dots , c_n) \in {\mathcal {SS}}_{n-1,k,s-1}^{(r)}$, one sees that $(a, c_2, \dots , c_n) \in {\mathcal {SS}}_{n,k,s}^{(r)}$ by prepending an $s-1$ to any $(n-1,k,s-1)$-staircase $(b_2, \dots , b_n) \geq (c_2, \dots , c_n)$.

• • Finally, if $c_1 = a$ is unbarred and $c_1 \geq s$, the first letter $b_1$ of $(b_1, b_2, \dots , b_n)$ must be unbarred and $\geq s$. This implies that $b_1 = k-1$, so that $s \leq a < k-1$. Furthermore, the sequence $(b_2, \dots , b_n)$ is a shuffle of the words

  $$ \begin{align*} ( \overline{k-s-1} )^{r_0} (s-1) ( \overline{k-s} )^{r_1} (s-2) ( \overline{k-s+1} )^{r_2} (s-3) \cdots 0 ( \overline{k-1} )^{r_s} \quad \text{and} \quad (k-1)^{n-r-s-1}, \end{align*} $$
  where $r_0 + r_1 + \cdots + r_s = r$. This means that $(c_2, \dots , c_n) \in {\mathcal {SS}}_{n-1,k,s}^{(r)}$. Conversely, for any $(c_2, \dots , c_n) \in {\mathcal {SS}}_{n-1,k,s}^{(r)}$, we see that $(a, c_2, \dots , c_n) \in {\mathcal {SS}}_{n,k,s}^{(r)}$ by prepending a $k-1$ to any $(n-1,k,s)$-staircase $(b_2, \dots , b_n) \geq (c_2, \dots , c_n)$.

The three bullet points above show that ${\mathcal {SS}}_{n,k,s}^{(r)}$ is a union of the claimed sets of words. The disjointness of this union follows since the first letters of the words in these sets are distinct.

We are ready to state the main result of this subsection: the codes of ordered set superpartitions are precisely the substaircase words. The key idea of the proof is to invert the map ${\mathrm {code}}: \sigma \mapsto (c_1, \dots , c_n)$, sending an ordered set partition to its coinversion code. The inverse map $\iota : (c_1, \dots , c_n) \mapsto \sigma $ is a variant on the insertion maps in the proof of Theorem 3.8.

Theorem 3.13. Let $n, k, s \geq 0$ be integers with $k \geq s$, and let $r \leq n-k$. The coinversion code map gives a well-defined bijection

$$ \begin{align*} {\mathrm {code}}: {\mathcal{OSP}}_{n,k,s}^{(r)} \longrightarrow {\mathcal{SS}}_{n,k,s}^{(r)}. \end{align*} $$

Proof. Our first task is to show that the function ${\mathrm {code}}$ is well-defined, that is that ${\mathrm {code}}(\sigma ) \in {\mathcal {SS}}_{n,k,s}^{(r)}$ for any $\sigma \in {\mathcal {OSP}}_{n,k,s}^{(r)}$. To this end, let $\sigma = (B_1 \mid \cdots \mid B_k) \in {\mathcal {OSP}}_{n,k,s}^{(r)}$ with ${\mathrm {code}}(\sigma ) = (c_1, \dots , c_n)$. We associate a staircase $(b_1, \dots , b_n)$ to $\sigma $ as follows. Write the minimal elements $\min B_1, \dots , \min B_s$ in increasing order $i_1 < \cdots < i_s$, and set $b_{i_j} = s-j$ for each $j \in \{1,\ldots ,s\}$. If $1 \leq i \leq n$ and i is barred in $\sigma $, write $b_i = \overline {k-s-1+m_i}$, where $m_i = | \{ 1 \leq j \leq s \,:\, i_j < i \} |$. Finally, if $1 \leq i \leq n$, i is unbarred in $\sigma $, and i is not minimal in any of the first s blocks $B_1, \dots , B_s$, set $b_i = k-1$.

As an example of these concepts, consider $\sigma = (5, 7 \mid 1 \mid 3, \bar {4}, \bar {8} \mid \varnothing \mid \bar {2}, 6) \in {\mathcal {OSP}}_{8,5,3}^{(3)}$. The associated staircase is $(b_1, \dots , b_8) = (2, \bar {2}, 1, \bar {3}, 0, 4, 4, \bar {4})$. We have ${\mathrm {code}}(\sigma ) = (c_1, \dots , c_8) = (1, \bar {2}, 0, \bar {1}, 0, 4, 0, \bar {2})$, which has the same bar pattern as, and is componentwise $\leq $, the sequence $(b_1, \dots , b_8)$.

By construction, b is indeed a staircase. Furthermore, $b_i$ is barred if and only if $c_i$ is barred. It is also the case that $c_i \leq b_i$ for $1 \leq i \leq n$. One way to see this is to note that b is the unique set superpartition $\tau $ that has the same positive-height entries, the same zero-height entries and the same negative-height entries as $\sigma $ but with its code maximised in every coordinate. We can construct $\tau $ by writing the zero-height entries in increasing order from left to right and placing the other entries as far right as possible. In our example, we have $\tau = (1 \mid 3 \mid 5 \mid \varnothing \mid \bar {2}, \bar {4}, 6, 7, \bar {8}) \in {\mathcal {OSP}}_{8,5,3}^{(3)}$.

We leave it for the reader to verify that $(c_1, \dots , c_n)$ has the same bar pattern as $(b_1, \dots , b_n)$ and $(c_1, \dots , c_n) \leq (b_1, \dots , b_n)$ componentwise. This shows that the function ${\mathrm {code}}$ in the statement is well defined.

We show that ${\mathrm {code}}$ is a bijection by constructing its inverse map

(3.22)$$ \begin{align} \iota: {\mathcal{SS}}_{n,k,s}^{(r)} \longrightarrow {\mathcal{OSP}}_{n,k,s}^{(r)}\\[-16pt]\nonumber \end{align} $$

as follows. The map $\iota $ starts with a sequence $(\varnothing \mid \cdots \mid \varnothing )$ of k copies of the empty set and builds up an ordered set superpartition by insertion.

Given a sequence $(B_1 \mid \cdots \mid B_k)$ of k possibly empty subsets of the alphabet $\{1, 2, \dots , \bar {1}, \bar {2}, \dots \}$, we assign the blocks $B_i$ the unbarred labels $0, 1, 2, \dots , k-1$ as follows. Moving from right to left, we assign the labels $0, 1, \dots , j-1$ to the j empty blocks among $B_1, \dots , B_s$. Then, moving from left to right, we assign the unlabeled blocks the labels $j, j+1, \dots , k-1$. An example of unbarred labels when $k = 7$ and $s = 4$ is as follows:

$$ \begin{align*} ( 3, \bar{4} \,_2 \mid \varnothing \,_1 \mid \varnothing \,_0 \mid 1, 5 \,_3 \mid \varnothing \,_4 \mid \bar{2} \,_5 \mid \varnothing \,_6 );\\[-16pt] \end{align*} $$

we draw unbarred labels below their blocks. The barred labels $\bar {0}, \bar {1}, \dots $, as assigned to the blocks $B_i$, where either $i> s$ or $B_i \neq \varnothing $ by moving left to right. In our example, the barred labels are

$$ \begin{align*} ( 3, \bar{4} \,^{\bar{0}} \mid \varnothing \mid \varnothing \mid 1, 5 \,^{\bar{1}} \mid \varnothing \,^{\bar{2}} \mid \bar{2} \,^{\bar{3}} \mid \varnothing \,^{\bar{4}} ),\\[-16pt] \end{align*} $$

where the blocks $B_2 = B_3 = \varnothing $ do not receive a barred label because $2, 3 \leq s = 4$. Barred labels are written above their blocks.

Let $(c_1, \dots , c_n) \in {\mathcal {SS}}_{n,k,s}^{(r)}$. To define $\iota (c_1, \dots , c_n)$, we start with the sequence $(B_1 \mid \cdots \mid B_k) = (\varnothing \mid \cdots \mid \varnothing )$ of k empty blocks and iteratively insert i into the block with unbarred label $c_i$ (if $c_i$ is unbarred) or insert $\bar {i}$ into the block with barred label $c_i$ (if $c_i$