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In 1974, Carlitz and Scoville introduced the Stirling–Eulerian polynomial A n (x , y | α , β) as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of A n (x , y | α , β) , denoted P n (u 1 , u 2 , u 3 , u 4 | α , β) , which is the enumerator of permutations by valleys, peaks, double ascents, double descents, left-to-right maxima and right-to-left maxima. Using Chen's context-free grammar calculus, Ji proved a formula for the generating function of P n (u 1 , u 2 , u 3 , u 4 | α , β) , generalizing the work of Carlitz and Scoville. Ji's formula has many nice consequences, one of which is an intriguing γ -positivity expansion for A n (x , y | α , β) . In this paper, we prove a q-analog of Ji's formula by using Gessel's q-compositional formula and provide a combinatorial approach to her γ -positivity expansion of A n (x , y | α , β) . [ABSTRACT FROM AUTHOR]

There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r + 1 . Motivated by work on rational numbers in the Calkin–Wilf tree, we classify partitions whose corresponding binary sequence is a palindrome. We give a generating function that counts these partitions, and describe how to efficiently generate all of them. Atypically for partition generating functions, we find an unusual significance to prime degrees. Specifically, we prove there are nontrivial palindrome partitions of n except when n = 3 or n + 1 is prime. We find an interesting new "branching diagram" for partitions, similar to Young's lattice, with an action of the Klein four group corresponding to natural operations on the binary sequences. [ABSTRACT FROM AUTHOR]

A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index i is either a cycle valley ( σ - 1 (i) > i < σ (i) ) or a cycle peak ( σ - 1 (i) < i > σ (i) ). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes–Rogers and Jacobi–Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs. [ABSTRACT FROM AUTHOR]

We present what we call a "motivated proof" of the Bressoud-Göllnitz-Gordon identities. Similar "motivated proofs" have been given by Andrews and Baxter for the Rogers–Ramanujan identities and by Lepowsky and Zhu for Gordon's identities. Additionally, "motivated proofs" have also been given for the Andrews-Bressoud identities by Kanade, Lepowsky, Russell, and Sills and for the Göllnitz–Gordon–Andrews identities by Coulson, Kanade, Lepowsky, McRae, Qi, Russell, and the third author. Our proof borrows both the use of "ghost series" from the "motivated proof" of the Andrews–Bressoud identities and uses recursions similar to those found in the "motivated proof" of the Göllnitz–Gordon–Andrews identities. We anticipate that this "motivated proof" of the Bressoud–Göllnitz–Gordon identities will illuminate certain twisted vertex-algebraic constructions. [ABSTRACT FROM AUTHOR]

Chan and Cooper proved that if the integers c (n) (n = 0 , 1 , 2 , ...) are given by ∑ n = 0 ∞ c (n) q n = ∏ n = 1 ∞ 1 1 - q n 2 1 - q 3 n 2 , then ∑ n = 0 ∞ c (2 n + 1) q n = 2 ∏ n = 1 ∞ 1 - q 2 n 4 1 - q 6 n 4 1 - q n 6 1 - q 3 n 6 . We prove many other results of this type and apply them to the determination of congruence properties of the coefficients. [ABSTRACT FROM AUTHOR]

Munarini (J Integer Seq 23: Article 20.3.8, 2020) recently showed that the derangement polynomial d n (q) = ∑ σ ∈ D n q maj (σ) is expressible as the determinant of either an n × n tridiagonal matrix or an n × n lower Hessenberg matrix. Qi et al. (Cogent Math 3:1232878, 2016) showed that the classical derangement number d n = n ! ∑ k = 0 n (- 1) k k ! is expressible as a tridiagonal determinant of order n + 1 . We show in this work that similar determinantal expressions exist for the type B derangement polynomial d n B (q) = ∑ σ ∈ D n B q fmaj (σ) studied previously by Chow (Sém Lothar Combin 55:B55b, 2006), and the type D derangement polynomial d n D (q) = ∑ σ ∈ D n D q maj (σ) studied recently by Chow (Taiwanese J Math 27(4):629–646, 2023). Representations of the types B and D derangement numbers d n B and d n D as determinants of order n + 1 are also presented. [ABSTRACT FROM AUTHOR]

In 2018, Lin and Wang introduced two restricted partition triples, whose generating functions are related to the reciprocals of Ramanujan–Gordon identities. Let R G 2 (n) denote the number of partitions triples of n, where odd parts in the first two components are distinct and the last component only contains even parts. Lin and Wang proved some congruences modulo 5 and 7 satisfied by R G 2 (n) . They further asked the existence of the infinite families of congruences modulo high powers of 2 for the function R G 2 (n) . Utilizing some q-techniques, we prove that such congruence families indeed exist. [ABSTRACT FROM AUTHOR]

Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews–Crandall-type identity and use it to count the number of integer solutions to x 2 + 2 y 2 + 2 z 2 = n . [ABSTRACT FROM AUTHOR]

For integers a and b (not both 0) we define the integers c (a , b ; n) (n = 0 , 1 , 2 , ...) by ∑ n = 0 ∞ c (a , b ; n) q n = ∏ n = 1 ∞ 1 - q n a (1 - q 2 n) b (| q | < 1). These integers include the numbers t k (n) = c (- k , 2 k ; n) , which count the number of representations of n as a sum of k triangular numbers, and the numbers (- 1) n r k (n) = c (2 k , - k ; n) , where r k (n) counts the number of representations of n as a sum of k squares. A computer search was carried out for integers a and b, satisfying - 24 ≤ a , b ≤ 24 , such that at least one of the sums 0.1 ∑ n = 0 ∞ c (a , b ; 3 n + j) q n , j = 0 , 1 , 2 , is either zero or can be expressed as a nonzero constant multiple of the product of a power of q and a single infinite product of factors involving powers of 1 - q rn with r ∈ { 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24 } for all powers of q up to q 1000 . A total of 84 such candidate identities involving 56 pairs of integers (a, b) all satisfying a ≡ b (mod 3) were found and proved in a uniform manner. The proof of these identities is extended to establish general formulas for the sums (0.1). These formulas are used to determine formulas for the sums ∑ n = 0 ∞ t k (3 n + j) q n , ∑ n = 0 ∞ r k (3 n + j) q n , j = 0 , 1 , 2. [ABSTRACT FROM AUTHOR]

The minimal excludant, or mex-function, on a set of positive integers is the smallest positive integer not in it. Andrews and Newman defined the mex-function mex A , a (λ) to be the smallest positive integer congruent to a modulo A that is not part of partition λ , and denote by p A , a (n) (reps. p ¯ A , a (n) ) the number of partitions λ of n satisfying mex A , a (λ) ≡ a (reps. a + A) (mod 2 A) , and found numerous surprising identities involving these functions. Motivated by the above results, in this paper, we prove that the number of the partitions of n with an even (resp. odd) number of even parts equals the mex-function p 4 , 2 (n) (reps. p ¯ 4 , 2 (n) ). We also derive several identities connecting the differences of two mex-functions with partitions restricted by certain congruences, which develops the work of Dhar, Mukhopadhyay, and Sarma. Furthermore, we extend the mex-function to overpartitions and study the relevant properties. [ABSTRACT FROM AUTHOR]

In this paper, we study, for a given arithmetic function f, the sequence of polynomials (P n f (t)) n = 0 ∞ , defined by the recurrence P 0 f (x) = 1 , P 1 f (x) = x , P n f (x) = x n ∑ k = 1 n f (k) P n - k f (x) , n ≥ 2. Using the ideas from the paper by Heim, Luca, and Neuhauser, we prove, under some assumptions on P n f (t) for 1 ≤ n ≤ 10 , that no root of unity can be a root of any polynomial P n f (t) for n ∈ N . Then we specify the result to some functions f related to colored partitions. [ABSTRACT FROM AUTHOR]

For primes ℓ and nonnegative integers a, we study the partition functions p ℓ (a ; n) : = # { λ ⊢ n : ord ℓ (H (λ)) = a } , where H (λ) denotes the product of hook lengths of a partition λ . These partition values arise as the McKay numbers m ℓ (ord ℓ (n !) - a ; S n) in the representation theory of the symmetric group. We determine the generating functions for p ℓ (a ; n) in terms of p ℓ (0 ; n) and specializations of specific D'Arcais polynomials. For ℓ = 2 and 3, we give an exact formula for the p ℓ (a ; n) and prove that these values are zero for almost all n. For larger primes ℓ , the p ℓ (a ; n) are positive for sufficiently large n. Despite this positivity, we prove that p ℓ (a ; n) is almost always divisible by m for any integer m. Furthermore, with these results we prove several Ramanujan-type congruences. These include the congruences p ℓ (a ; ℓ k n - δ (a , ℓ)) ≡ 0 (mod ℓ k + 1) , for 0 < a < ℓ , where ℓ = 5 , 7 , 11 and δ (a , ℓ) : = (ℓ 2 - 1) / 24 + a ℓ , which answer a question of Ono. [ABSTRACT FROM AUTHOR]

Using Ramanujan sums, which generalize the Möbius function and the Euler totient function, we enumerate q-ary (fixed) necklaces over the color set { a 1 , ... , a q } with n i beads of color a i for i = 1 , ... , q and co-periods dividing a fixed non-negative integer v. (The co-period of a necklace is its length divided by its period.) We also provide Witt-type infinite products and Lambert-type multiple infinite series for these quantities. Furthermore, we provide regions in C q where our infinite products and series converge absolutely. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain some interesting infinite series representations for the Apostol-type parametrically generalized polynomials with the aid of the Laplace transform and generating functions. In particular, by using the method of generating functions, we derive not only recurrence relations, but also several other formulas, identities, and relations as well as combinatorial sums for these parametrically generalized numbers and polynomials and for other known special numbers and polynomials. These identities, relations and combinatorial sums are related to the two-parameter types of the Apostol–Bernoulli polynomials of higher order, the two-parameter types of Apostol–Euler polynomials of higher order, the two-parameter types of Apostol–Genocchi polynomials of higher order, the Apostol–Bernoulli polynomials of higher order, the Apostol–Euler polynomials of higher order, the Apostol–Genocchi polynomials of higher order, the cosine- and sine-Bernoulli polynomials, the cosine- and sine-Euler polynomials, the λ -array-type polynomials, the λ -Stirling numbers, the polynomials C n x , y , and the polynomials S n x , y . Finally, we present several new recurrence relations for these special polynomials and numbers. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of sequences in overpartitions and their relation to 2-color partitions. An extensive study of a general class of double series is required to achieve these ends. [ABSTRACT FROM AUTHOR]

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( q = 1 ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions and provide closed forms for the generating functions of these partitions. [ABSTRACT FROM AUTHOR]

By applying the partial derivative operator to a summation of hypergeometric series, we prove two double series for π , which were conjectured by Guo and Lian recently. Employing the operator just mentioned and a summation formula for basic hypergeometric series, we also give q-analogues of these two double series. [ABSTRACT FROM AUTHOR]

In his recent work Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions. In this note we provide bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors. [ABSTRACT FROM AUTHOR]

We introduce a new family of hyperplane arrangements inspired by the homogenized Linial arrangement (which was recently introduced by Hetyei), and show that the intersection lattices of these arrangements are isomorphic to the bond lattices of Ferrers graphs. Using recent work of Lazar and Wachs we are able to give combinatorial interpretations of the characteristic polynomials of these arrangements in terms of permutation enumeration. For certain infinite families of these hyperplane arrangements, we are able to give generating function formulas for their characteristic polynomials. To do so, we develop a generalization of Dumont's surjective staircases, and introduce a polynomial which enumerates these generalized surjective staircases according to several statistics. We prove a recurrence for these polynomials and show that in certain special cases this recurrence can be solved explicitly to yield a generating function. We also prove refined versions of several of these results using the theory of complex hyperplane arrangements. [ABSTRACT FROM AUTHOR]

In 2008, Muneta found explicit evaluation of the multiple zeta star value ζ ⋆ ({ 3 , 1 } d) , and in 2013, Yamamoto proved a sum formula for multiple zeta star values on 3–2–1 indices. In this paper, we provide another way of deriving the formulas mentioned above. It is based on our previous work on generating functions for multiple zeta star values and also on constructions of generating functions for restricted sums of alternating Euler sums. As a result, the formulas obtained are simpler and computationally more effective than the known ones. Moreover, we give explicit evaluations of ζ ⋆ ({ { 2 } m , 3 , { 2 } m , 1 } d) and ζ ⋆ ({ { 2 } m , 3 , { 2 } m , 1 } d , { 2 } m + 1) , which are new and have not appeared in the literature before. [ABSTRACT FROM AUTHOR]

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form ∏ j = 1 n ∏ k = 1 p - 1 (1 - q p j - k) s , where p is an odd prime and n, s are positive integers. Precisely, let a i denote the coefficient of q i in the above polynomial and suppose that b is an integer. We prove that | ∑ i ≡ b mod 2 p n a i - v (b) p sn 2 p n | ≤ p s n / 2 , where v (b) = p - 1 if b divisible by p and v (b) = - 1 otherwise. This improves a recent result of Goswami and Pantangi (Ramanujan J, 2021. https://doi.org/10.1007/s11139-020-00352-0). [ABSTRACT FROM AUTHOR]

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations, called alternating Eulerian polynomials, are unimodal via a five-term recurrence relation. We also find a quadratic recursion for the alternating major index q-analog of the alternating Eulerian polynomials. As an interesting application of this quadratic recursion, we show that (1 + q) ⌊ n / 2 ⌋ divides ∑ π ∈ S n q altmaj (π) , where S n is the set of all permutations of { 1 , 2 , ... , n } and altmaj (π) is the alternating major index of π . This leads us to discover a q-analog of n ! = 2 ℓ m , m odd, using the statistic of alternating major index. Moreover, we study the γ -vectors of the alternating Eulerian polynomials by using these two recursions and the cd-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work. [ABSTRACT FROM AUTHOR]

We study the number of ordered k-partitions of a multiset with equal sums, having elements α 1 , ... , α n and multiplicities m 1 , ... , m n . Denoting this number by S k (α 1 , ... , α n ; m 1 , ... , m n) , we find the generating function, derive an integral formula, and illustrate the results by numerical examples. The special case involving the set { 1 , ⋯ , n } presents particular interest and leads to the new integer sequences S k (n) , Q k (n) , and R k (n) , for which we provide explicit formulae and combinatorial interpretations. Conjectures in connection to some superelliptic Diophantine equations and an asymptotic formula are also discussed. The results extend previous work concerning 2- and 3-partitions of multisets. [ABSTRACT FROM AUTHOR]

A partition α is said to contain another partition (or pattern) μ if the Ferrers board for μ is attainable from α under removal of rows and columns. We say α avoids μ if it does not contain μ . In this paper we count the number of partitions of n avoiding a fixed pattern μ , in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever μ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic p-groups. We further obtain asymptotics for the number of partitions of n avoiding a pattern μ . Using these asymptotics we conclude that the generating function for μ is not algebraic whenever μ is rook equivalent to a partition with distinct parts whose first two parts are positive and differ by 1. [ABSTRACT FROM AUTHOR]

We are concerned with counting self-conjugate (s , s + 1 , s + 2) -core partitions. A Motzkin path of length n is a path from (0, 0) to (n, 0) which stays weakly above the x-axis and consists of the up U = (1 , 1) , down D = (1 , - 1) , and flat F = (1 , 0) steps. We say that a Motzkin path of length n is symmetric if its reflection about the line x = n / 2 is itself. In this paper, we show that the number of self-conjugate (s , s + 1 , s + 2) -cores is equal to the number of symmetric Motzkin paths of length s, and give a closed formula for this number. [ABSTRACT FROM AUTHOR]

In this paper, we expand on the work of Guo and Zeng (Adv Appl Math 39(1):96–115, 2007) on a generalisation of the Ramanujan polynomials and planar trees. We manage to find combinatorial interpretations of this family of polynomials in terms of Greg trees, Cayley trees and planar trees by constructing bijections that preserve relevant tree statistics. [ABSTRACT FROM AUTHOR]

Let p k (n) be given by the series expansion of the k-th power of the Euler Product ∏ n = 1 ∞ (1 - q n) k = ∑ n = 0 ∞ p k (n) q n . By investigating the properties of the modular equations of the second and the third order under the Atkin U-operator, we determine the generating functions of p 8 k (2 2 α n + k (2 2 α - 1) 3) (1 ≤ k ≤ 3) and p 3 k (3 2 β n + k (3 2 β - 1) 8) (1 ≤ k ≤ 8) in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo m, we obtain infinite families of congruences for p k (n) modulo any m ≥ 2 , where 1 ≤ k ≤ 24 and 3|k or 8|k. Based on these congruences for p k (n) , infinite families of congruences for many partition functions such as the overpartition function, t-core partition functions and ℓ -regular partition functions are easily obtained. [ABSTRACT FROM AUTHOR]

Cellular automata are discrete dynamical systems that consist of patterns of symbols on a grid, which change according to a locally determined transition rule. In this paper, we will consider cellular automata that arise from polynomial transition rules, where the symbols are integers modulo some prime p. We consider the asymptotic behavior of the line complexity sequenceaT(k), which counts, for each k, the number of coefficient strings of length k that occur in the automaton. We begin with the modulo 2 case. For a polynomial T(x)=c0+c1x+⋯+cnxn with c0,cn≠0, we construct odd and even parts of the polynomial from the strings 0c1c3c5⋯c1+2⌊(n-1)/2⌋ and c0c2c4⋯c2⌊n/2⌋, respectively. We prove that aT(k) satisfies recursions of a specific form if the odd and even parts of T are relatively prime. We also define the order of such a recursion and show that the property of “having a recursion of some order” is preserved when the transition rule is raised to a positive integer power. Extending to a more general setting, we consider an abstract generating function ϕ(z)=∑k=1∞α(k)zk which satisfies a functional equation relating ϕ(z) and ϕ(zp). We show that there is a continuous, piecewise quadratic function f on [1 / p, 1] for which limk→∞(α(k)/k2-f(p-⟨logpk⟩))=0 (here ⟨y⟩=y-⌊y⌋). We use this result to show that for certain positive integer sequences sk(x)→∞ with a parameter x∈[1/p,1], the ratio α(sk(x))/sk(x)2 tends to f(x), and that the limit superior and inferior of α(k)/k2 are given by the extremal values of f. [ABSTRACT FROM AUTHOR]

Let D be a subset of a finite commutative ring R with identity. Let f(x)∈R[x] be a polynomial of degree d. For a nonnegative integer k, we study the number Nf(D,k,b) of k-subsets S in D such that ∑x∈Sf(x)=b.In this paper, we establish several bounds for the difference between Nf(D,k,b) and the expected main term 1|R||D|k, depending on the nature of the finite ring R and f. For R=Zn, let p=p(n) be the smallest prime divisor of n, |D|=n-c≥Cdnp-1d+c and f(x)=adxd+⋯+a0∈Z[x] with (ad,…,a1,n)=1. Then Nf(D,k,b)-1nn-ck≤δ(n)(n-c)+(1-δ(n))Cdnp-1d+c+k-1k,answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where δ(n)=∑i∣n,μ(i)=-11i and Cd=e1.85d. Furthermore, if n is a prime power, then δ(n)=1/p and one can take Cd=4.41. Similar and stronger bounds are given for two more cases. The first one is when R=Fq, a q-element finite field of characteristic p and f(x) is general. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results. [ABSTRACT FROM AUTHOR]

Let bk(n) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for b7(n) and b49(n). For example, for all j≥1 and n≥0, we prove that b7(72j-1n+3·72j-1-14)≡0(mod7j)and b49(7jn+7j-2)≡0(mod7j). [ABSTRACT FROM AUTHOR]

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled “Square Series Generating Function Transformations” (arXiv:1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of qn2 for some fixed non-zero q with |q|<1, we study the expansions of these special series through power series generated by Jacobi-type continued fractions, or J-fractions. We prove new exact expansions of the hth convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists as h→∞. We also prove new infinite q-series representations of special square series expansions involving square-power terms of the series parameter q, the q-Pochhammer symbol, and double sums over the q-binomial coefficients. Applications of the new results we prove within the article include new q-series representations for the ordinary generating functions of the special sequences, rp(n), and σ1(n), as well as parallels to the examples of the new integral representations for theta functions, series expansions of infinite products and partition function generating functions, and related unilateral special function series cited in the first square series transformations article. [ABSTRACT FROM AUTHOR]

In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is n is equal to the number of partitions of n with non-negative crank. [ABSTRACT FROM AUTHOR]

Stirling numbers of both kinds are linked to each other via two combinatorial identities due to Schläfli and Gould. Using q-analogs of Stirling numbers defined as inversion generating functions, we provide q-analogs of the two identities. The proof is computational and we leave open the problem of finding a more combinatorial one. [ABSTRACT FROM AUTHOR]

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theorem in this setting, and connect it with the Rogers-Fine identity. We conclude with some congruence properties. [ABSTRACT FROM AUTHOR]

The concept of t-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of t-hooks. It is well known that the hook lengths of multiples of t can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes modulo t. [ABSTRACT FROM AUTHOR]

We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle. [ABSTRACT FROM AUTHOR]

Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the generating function for Young books according to major index statistic is considered. It is shown that this generating function can be written as a Jackson integral which gives a new q-Selberg integral. It is also shown that the new q-Selberg integral has an expression in terms of Schur functions. [ABSTRACT FROM AUTHOR]

We present what we call a 'motivated proof' of the Göllnitz-Gordon-Andrews identities. A similar motivated proof of the Rogers-Ramanujan identities was previously given by G. E. Andrews and R. J. Baxter, and was subsequently generalized to Gordon's identities by J. Lepowsky and M. Zhu. We anticipate that the present proof of the Göllnitz-Gordon-Andrews identities will illuminate certain twisted vertex-algebraic constructions. [ABSTRACT FROM AUTHOR]

Fourier-Dedekind sums are a generalization of Dedekind sums-important number-theoretical objects that arise in many areas of mathematics, including lattice point enumeration, signature defects of manifolds, and pseudorandom number generators. A remarkable feature of Fourier-Dedekind sums is that they satisfy a reciprocity law called Rademacher reciprocity. In this paper, we study several aspects of Fourier-Dedekind sums: properties of general Fourier-Dedekind sums, extensions of the reciprocity law, average behavior of Fourier-Dedekind sums, and finally, extrema of 2-dimensional Fourier-Dedekind sums. On properties of general Fourier-Dedekind sums, we show that a general Fourier-Dedekind sum is simultaneously a convolution of simpler Fourier-Dedekind sums, and, in a precise sense, almost a linear combination of these with integer coefficients. We show that Fourier-Dedekind sums can be extended naturally to a group under convolution. We introduce 'Reduced Fourier-Dedekind sums,' which encapsulate the complexity of a Fourier-Dedekind sum, describe these in terms of generating functions, and give a geometric interpretation. Next, by finding interrelations among Fourier-Dedekind sums, we extend the range on which Rademacher Reciprocity Theorem holds. We study the average behavior of Fourier-Dedekind sums, showing that the average behavior of a Fourier-Dedekind sum is described concisely by a lower-dimensional, simpler Fourier-Dedekind sum. Finally, we focus our study on 2-dimensional Fourier-Dedekind sums. We find tight upper and lower bounds on these for a fixed $$t$$ , estimates on the argmax and argmin, and bounds on the sum of their 'reciprocals.' [ABSTRACT FROM AUTHOR]

We present a new companion to the deep partition theorem of Göllnitz and discuss it in the context of a generalization of Göllnitz's theorem by Alladi-Andrews-Gordon that was obtained by the method of weighted words. After providing a q-theoretic proof of the new companion theorem, we discuss its analytic representation and its link to the key identity of Alladi-Andrews-Gordon. [ABSTRACT FROM AUTHOR]

Let $$\mathcal {T}$$ be a collection of 3-element subsets $$S$$ of $$\{1,\dots ,n\}$$ with the property that if $$i

In our previous paper (J. Comb. Theory Ser. A 120(1):28–38, 2013 ), we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the Schröter, Russell, and Ramanujan type. The goal of this paper is to use the master bijection of Sandon and Zanello (J. Comb. Theory Ser. A 120(1):28–38, 2013 ) to show combinatorially several new and highly nontrivial colored partition identities. We conclude by listing a number of further interesting identities of the same type as conjectures. [ABSTRACT FROM AUTHOR]