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We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.
Andrews [‘Binary and semi-Fibonacci partitions’, J. Ramanujan Soc. Math. Math. Sci.7(1) (2019), 1–6] recently proved a new identity between the cardinalities of the set of semi-Fibonacci partitions and the set of partitions into powers of 2 with all parts appearing an odd number of times. We extend the identity to the set of semi-
$m$
-Fibonacci partitions of
$n$
and the set of partitions of
$n$
into powers of
$m$
in which all parts appear with multiplicity not divisible by
$m$
. We also give a new characterisation of semi-
$m$
-Fibonacci partitions and some congruences satisfied by the associated number sequence.
Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano–Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the existence of a lower bound for the growth rate independent of the finite generating set and the infinite index quasi-convex subgroup.
We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous
$\text{C}^{\ast }$
-algebras with exactly
$n$
Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.
Let
$n,r,k\in \mathbb{N}$
. An
$r$
-colouring of the vertices of a regular
$n$
-gon is any mapping
$\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$
. Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An
$r$
-ary necklace of length
$n$
is an equivalence class of
$r$
-colourings of
$\mathbb{Z}_{n}$
. We say that a colouring is
$k$
-alternating if all
$k$
consecutive vertices have pairwise distinct colours. We compute the smallest number
$r$
for which there exists a
$k$
-alternating
$r$
-colouring of
$\mathbb{Z}_{n}$
and we count, for any
$r$
, 2-alternating
$r$
-colourings of
$\mathbb{Z}_{n}$
and 2-alternating
$r$
-ary necklaces of length
$n$
.
We show that for any n and q, the number of real conjugacy classes in $ \rm{PGL}(\it{n},\mathbb{F}_q) $ is equal to the number of real conjugacy classes of $ \rm{GL}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SL}(\it{n},\mathbb{F}_q) $, refining a result of Lehrer [J. Algebra36(2) (1975), 278–286] and extending the result of Gill and Singh [J. Group Theory14(3) (2011), 461–489] that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $ \rm{PGU}(\it{n},\mathbb{F}_q) $, and equal to the number of real conjugacy classes of $ \rm{U}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SU}(\it{n},\mathbb{F}_q) $, refining results of Gow [Linear Algebra Appl.41 (1981), 175–181] and Macdonald [Bull. Austral. Math. Soc.23(1) (1981), 23–48]. We also give a generating function for this common quantity.
Fixing a positive integer r and
$0 \les k \les r-1$
, define
$f^{\langle r,k \rangle }$
for every formal power series f as
$ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$
Jochemko recently showed that the polynomial
$U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$
has only non-positive zeros for any
$r \ges \deg h(x) -k$
and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial
$h(x)$
of a lattice polytope of dimension n, which states that
$U^{n}_{r,0}\,h(x)$
has only negative, real zeros whenever
$r\ges n$
. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence
$( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$
is interlacing, so is
$( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$
. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.
the pioneer of interchange laws in universal algebra
We establish a combinatorial model for the Boardman–Vogt tensor product of several absolutely free operads, that is, free symmetric operads that are also free as 𝕊-modules. Our results imply that such a tensor product is always a free 𝕊-module, in contrast with the results of Kock and Bremner–Madariaga on hidden commutativity for the Boardman–Vogt tensor square of the operad of non-unital associative algebras.
We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the
$h^{\ast }$
-polynomial in case of complete bipartite graphs. In particular, we show that the
$h^{\ast }$
-polynomial is
$\unicode[STIX]{x1D6FE}$
-positive and real-rooted. This proves Gal’s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On
$\unicode[STIX]{x1D6FE}$
-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].
We give the generating function of split
$(n+t)$
-colour partitions and obtain an analogue of Euler’s identity for split
$n$
-colour partitions. We derive a combinatorial relation between the number of restricted split
$n$
-colour partitions and the function
$\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$
. We introduce a new class of split perfect partitions with
$d(a)$
copies of each part
$a$
and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’, Indian J. Pure Appl. Math22(9) (1991), 737–743].
Boij–Söderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with Sam, extending the theory to the setting of
$\text{GL}_{k}$
-equivariant modules and sheaves on Grassmannians. Algebraically, we study modules over a polynomial ring in
$kn$
variables, thought of as the entries of a
$k\times n$
matrix. We give equivariant analogs of two important features of the ordinary theory: the Herzog–Kühl equations and the pairing between Betti and cohomology tables. As a necessary step, we also extend previous results, concerning the base case of square matrices, to cover complexes other than free resolutions. Our statements specialize to those of ordinary Boij–Söderberg theory when
$k=1$
. Our proof of the equivariant pairing gives a new proof in the graded setting: it relies on finding perfect matchings on certain graphs associated to Betti tables and to spectral sequences. As an application, we construct three families of extremal rays on the Betti cone for
$2\times 3$
matrices.
We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures.
Athanasiadis [‘A survey of subdivisions and local
$h$
-vectors’, in The Mathematical Legacy of Richard P. Stanley (American Mathematical Society, Providence, RI, 2017), 39–51] asked whether the local
$h$
-polynomials of type
$A$
cluster subdivisions have only real zeros. We confirm this conjecture and prove that the local
$h$
-polynomials for all the Cartan–Killing types have only real roots. Our proofs use multiplier sequences and Chebyshev polynomials of the second kind.
We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups
$\text{SO}^{\pm }(n,q)$
and
$\unicode[STIX]{x1D6FA}^{\pm }(n,q)$
. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group
$G$
is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups
$G$
other than the groups
$\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$
with
$q$
even. We prove computationally that for small
$m$
this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups
$\text{SO}^{\pm }(4m+2,q)$
with
$q$
odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in
$\text{SO}^{\pm }(n,q)$
and
$\unicode[STIX]{x1D6FA}^{\pm }(n,q)$
for all
$q$
, and we use these to compute the asymptotic behavior for the number of involutions in these groups when
$q$
is fixed and
$n$
grows.
A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {ck}. Notoriously hard to compute, the exact fractions ck have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c4 and c5 as well; both are ratios of enormous integers, the denominator of c5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of ck is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.
In this paper a determinant identity is established, from which a simple proof of the multivariate Lagrange–Good inversion formula follows directly. Further discussion on a discrete analogue of the Lagrange–Good inversion formula is also presented.
In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.
We apply the concept of braiding sequences to link polynomials to show polynomial growth bounds on the derivatives of the Jones polynomial evaluated on S1 and of the Brandt–Lickorish–Millett–Ho polynomial evaluated on [–2, 2] on alternating and positive knots of given genus. For positive links, boundedness criteria for the coefficients of the Jones, HOMFLY and Kauffman polynomials are derived. (This is a continuation of the paper ‘Applications of braiding sequences. I’: Commun. Contemp. Math.12(5) (2010), 681–726.)
We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to consider the case of affine cones
$s+\mathfrak{c}$
, where
$s$
is an arbitrary real vertex and
$\mathfrak{c}$
is a rational polyhedral cone. For a given rational subspace
$L$
, we define the intermediate generating functions
$S^{L}(s+\mathfrak{c})(\unicode[STIX]{x1D709})$
by integrating an exponential function over all lattice slices of the affine cone
$s+\mathfrak{c}$
parallel to the subspace
$L$
and summing up the integrals. We expose the bidegree structure in parameters
$s$
and
$\unicode[STIX]{x1D709}$
, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math.12 (2012), 435–469] and [Intermediate sums on polyhedra: computation and real Ehrhart theory. Mathematika59 (2013), 1–22]. The bidegree structure is key to a new proof for the Baldoni–Berline–Vergne approximation theorem for discrete generating functions [Local Euler–Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. Contemp. Math.452 (2008), 15–33], using the Fourier analysis with respect to the parameter
$s$
and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.
We prove the unimodality of some coloured
$q$
-Eulerian polynomials, which involve the flag excedances, the major index and the fixed points on coloured permutation groups, via two recurrence formulas. In particular, we confirm a recent conjecture of Mongelli about the unimodality of the flag excedances over type B derangements. Furthermore, we find the coloured version of Gessel’s hook factorisation, which enables us to interpret these two recurrences combinatorially. We also provide a combinatorial proof of a symmetric and unimodal expansion for the coloured derangement polynomial, which was first established by Shin and Zeng using continued fractions.