We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let
$p_{j,k,m} (n)$
be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for
$m \geq 2$
. We prove that
$p_{1,0,m} (n)$
is in general larger than
$p_{0,1,m} (n)$
. We also obtain asymptotic formulas for
$p_{1,0,m}(n)$
and
$p_{0,1,m}(n)$
for
$m \geq 2$
.
We provide a generalised Laplace expansion for the permanent function and, as a consequence, we re-prove a multinomial Vandermonde convolution. Some combinatorial identities are derived by applying special matrices to the expansion.
We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d1, …, dn) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes dj= λn + O(n1/2+ε) for some λ bounded away from 0 and 1.
In this paper, we show that the numbers of t-stack sortable n-permutations with k − 1 descents satisfy central and local limit theorems for t = 1, 2, n − 1 and n − 2. This result, in particular, gives an affirmative answer to Shapiro's question about the asymptotic normality of the Narayana numbers.
By taking square lattices as a two-dimensional analogue to Beatty sequences, we are motivated to define and explore the notion of complementary lattices. In particular, we present a continuous one-parameter family of complementary lattices. This main result then yields several novel examples of complementary sequences, along with a geometric proof of the fundamental property of Beatty sequences.
Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0,
We show that the Mallows measure on permutations of
$1,\dots ,n$
arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph
$K_{{\mathbb Z},{\mathbb Z}}$
falls into one of two classes: a countable family
$(\sigma _n)_{n\in {\mathbb Z}}$
of tame stable matchings, in which the length of the longest edge crossing k is
$O(\log |k|)$
as
$k\to \pm \infty $
, and an uncountable family of wild stable matchings, in which this length is
$\exp \Omega (k)$
as
$k\to +\infty $
. The tame stable matching
$\sigma _n$
has the law of the Mallows permutation of
${\mathbb Z}$
(as constructed by Gnedin and Olshanski) composed with the shift
$k\mapsto k+n$
. The permutation
$\sigma _{n+1}$
dominates
$\sigma _{n}$
pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.
We study random unlabelled k-trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle-pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov–Hausdorff–Prokhorov and Benjamini–Schramm limits that describe their asymptotic geometric shape on a global and local scale as the number of (k + 1)-cliques tends to infinity.
In this note, we evaluate sums of partial theta functions. Our main tool is an application of an extended version of the Bailey transform to an identity of Gasper and Rahman on $q$-hypergeometric series.
We discuss a truncated identity of Euler and present a combinatorial proof of it. We also derive two finite identities as corollaries. As an application, we establish two related
$q$
-congruences for sums of
$q$
-Catalan numbers, one of which has been proved by Tauraso [‘
$q$
-Analogs of some congruences involving Catalan numbers’, Adv. Appl. Math.48 (2012), 603–614] by a different method.
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.
For positive integers
$n$
and
$k$
, let
$r_{k}(n)$
denote the number of representations of
$n$
as a sum of
$k$
squares, where representations with different orders and different signs are counted as distinct. For a given positive integer
$m$
, by means of some properties of binomial coefficients, we derive some infinite families of congruences for
$r_{k}(n)$
modulo
$2^{m}$
. Furthermore, in view of these arithmetic properties of
$r_{k}(n)$
, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.
Let
$X$
be a nonempty set and
${\mathcal{P}}(X)$
the power set of
$X$
. The aim of this paper is to identify the unital subrings of
${\mathcal{P}}(X)$
and to compute its cardinality when it is finite. It is proved that any topology
$\unicode[STIX]{x1D70F}$
on
$X$
such that
$\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$
, where
$\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$
, is a unital subring of
${\mathcal{P}}(X)$
. It is also shown that
$X$
is finite if and only if any unital subring of
${\mathcal{P}}(X)$
is a topology
$\unicode[STIX]{x1D70F}$
on
$X$
such that
$\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$
if and only if the set of unital subrings of
${\mathcal{P}}(X)$
is finite. As a consequence, if
$X$
is finite with cardinality
$n\geq 2$
, then the number of unital subrings of
${\mathcal{P}}(X)$
is equal to the
$n$
th Bell number and the supremum of the lengths of chains of unital subalgebras of
${\mathcal{P}}(X)$
is equal to
$n-1$
.
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.
In this paper, we investigate
$\unicode[STIX]{x1D70B}(m,n)$
, the number of partitions of the bipartite number
$(m,n)$
into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between
$\unicode[STIX]{x1D70B}(m,n)$
and the crank statistic
$M(m,n)$
for integer partitions. Using this relation, we establish some uniform asymptotic formulas for
$\unicode[STIX]{x1D70B}(m,n)$
.
Recently, Brietzke, Silva and Sellers [‘Congruences related to an eighth order mock theta function of Gordon and McIntosh’, J. Math. Anal. Appl.479 (2019), 62–89] studied the number
$v_{0}(n)$
of overpartitions of
$n$
into odd parts without gaps between the nonoverlined parts, whose generating function is related to the mock theta function
$V_{0}(q)$
of order 8. In this paper we first present a short proof of the 3-dissection for the generating function of
$v_{0}(2n)$
. Then we establish three congruences for
$v_{0}(n)$
along certain progressions which are subsequences of the integers
$4n+3$
.
Let
$\unicode[STIX]{x1D707}(m,n)$
(respectively,
$\unicode[STIX]{x1D702}(m,n)$
) denote the number of odd-balanced unimodal sequences of size
$2n$
and rank
$m$
with even parts congruent to
$2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$
(respectively,
$0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$
) and odd parts at most half the peak. We prove that two-variable generating functions for
$\unicode[STIX]{x1D707}(m,n)$
and
$\unicode[STIX]{x1D702}(m,n)$
are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single
$C^{\infty }$
function in
$\mathbb{R}\times \mathbb{R}$
to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables
$w$
and
$q$
, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size
$2n$
with even parts congruent to
$0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$
and odd parts at most half the peak.
Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.
Let
$\overline{t}(n)$
be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for
$\overline{t}(n)$
modulo small powers of
$2$
and
$3$
have been established. We present two infinite families of congruences modulo
$5$
and
$27$
for
$\overline{t}(n)$
, the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, Proc. Math. Sci.129 (2019), Article 31].
We introduce a notion of
$q$
-deformed rational numbers and
$q$
-deformed continued fractions. A
$q$
-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the
$q$
-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the
$q$
-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties,
$q$
-deformation of the Farey graph, matrix presentations and
$q$
-continuants are given, as well as a relation to the Jones polynomial of rational knots.