arXiv subjects
Publications and source records attributed to Kathy Q. Ji.
This note is devoted to a combinatorial proof of a Schmidt type theorem due to Andrews and Paule. A four-variable refinement of Andrews and Paule's theorem is also obtained based on this combinatorial construction.
$q$-Kaplansky numbers were considered by Chen and Rota. We find that $q$-Kaplansky numbers are connected to the symmetric differences of Gaussian polynomials introduced by Reiner and Stanton. Based on the work of Reiner and Stanton, we establish the unimodality of $q$-Kaplansky numbers. We also show that $q$-Kaplansky numbers are the generating functions for the inversion number and the major index of two special kinds of $(0,1)$-sequences. Furthermore, we show that $q$-Kaplansky numbers are strongly $q$-log-concave.
We obtain an asymptotic formula for Andrews and Paule's broken $k$-diamond partition function $Δ_k(n)$ where $k=1$ or $2$. Based on this asymptotic formula, we derive that $Δ_k(n)$ satisfies the order $d$ Turán inequalities for $d\geq 1$ and for sufficiently large $n$ when $k=1$ and $ 2$ by using a general result of Griffin, Ono, Rolen and Zagier. We also show that Andrews and Paule's broken $k$-diamond partition function $Δ_k(n)$ is log-concave for $n\geq 1$ when $k=1$ and $2$. This leads to $Δ_k(a)Δ_k(b)\geΔ_k(a+b)$ for $a,b\ge 1$ when $k=1$ and $ 2$.
We introduce the cyclic major index of a cycle permutation and give a bivariate analogue of enumerative formula for the cyclic shuffles with a given cyclic descent numbers due to Adin, Gessel, Reiner and Roichman, which can be viewed as a cyclic analogue of Stanley's Shuffle Theorem. This gives an answer to a question of Adin, Gessel, Reiner and Roichman, which has been posed by Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.
In 1980, Bressoud conjectured a combinatorial identity $A_j=B_j$ for $j=0$ or $1$, where the function $A_j$ counts the number of partitions with certain congruence conditions and the function $B_j$ counts the number of partitions with certain difference conditions. Bressoud's conjecture specializes to a wide variety of well-known theorems in the theory of partitions. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim resolved Bressoud's conjecture for the case $j=1$. In this paper, we introduce a new partition function $\bar{B}_j$ which can be viewed as an overpartition analogue of the partition function $B_j$ introduced by Bressoud. By means of Gordon markings, we build bijections to obtain a relationship between $\bar{B}_1$ and $B_0$ and a relationship between $\bar{B}_0$ and $B_1$. Based on these former relationships, we further give overpartition analogues of many classical partition theorems including Euler's partition theorem, the Rogers-Ramanujan-Gordon identities, the Bressoud-Rogers-Ramanujan identities, the Andrews-Göllnitz-Gordon identities and the Bressoud-Göllnitz-Gordon identities.
The main objective of this paper is to investigate the distribution of the Andrews-Garvan-Dyson crank of a partition. Let $M(m,n)$ denote the number of partitions of $n$ with the Andrews-Garvan-Dyson crank $m$, we show that the sequence \break $\{M(m,n)\}_{|m|\leq n-1}$ is unimodal for $n\geq 44$. It turns out that the unimodality of \break $\{M(m,n)\}_{|m|\leq n-1}$ is related to the monotonicity properties of two partition \break functions $p_k(n)$ and $pp_k(n)$. Let $p_k(n)$ denote the number of partitions of $n$ with at most $k$ parts such that the largest part appears at least twice and let $pp_k(n)$ denote the number of pairs $(α,β)$ of partitions of $n$, where $α$ is a partition counted by $p_k(i)$ and $β$ is a partition counted by $p_{k+1}(n-i)$ for $0\leq i\leq n$. We show that $p_k(n)\geq p_k(n-1)$ for $k\geq 5$ and $n\geq 14$ and $pp_k(n)\geq pp_k(n-1)$ for $k\geq 3$ and $n\geq 2$. With the aid of the monotonicity properties on $p_k(n)$ and $pp_k(n)$, we show that $M(m,n)\geq M(m,n-1)$ for $n\geq 14$ and $ 0\leq m \leq n-2$ and $M(m-1,n)\geq M(m,n)$ for $n\geq 44$ and $1\leq m\leq n-1$. By means of the symmetry $M(m,n)=M(-m,n)$, we find that $M(m-1,n)\geq M(m,n)$ for $n\geq 44$ and $1\leq m\leq n-1$ implies that the sequence $\{M(m,n)\}_{|m|\leq n-1}$ is unimodal for $n\geq 44$. We also give a proof of an upper bound for ospt(n) conjectured by Chan and Mao in light of the inequality $M(m-1,n)\geq M(m,n)$ for $n\geq 44$ and $0\leq m\leq n-1$.
In 1967, Andrews found a combinatorial generalization of the Göllnitz-Gordon theorem, which can be called the Andrews-Göllnitz-Gordon theorem. In 1980, Bressoud derived a multisum Rogers-Ramanujan-type identity, which can be considered as the generating function counterpart of the Andrews-Göllnitz-Gordon theorem. Lovejoy gave an overpartition analogue of the Andrews-Göllnitz-Gordon theorem for $i=k$. In this paper, we give an overpartition analogue of this theorem in the general case. By using Bailey's lemma and a change of base formula due to Bressoud, Ismail and Stanton, we obtain an overpartition analogue of Bressoud's identity. We then give a combinatorial interpretation of this identity by introducing the Göllnitz-Gordon marking of an overpartition, which yields an overpartition analogue of the Andrews-Göllnitz-Gordon theorem.
Let $N(\leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(\leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(\leq m,n)\leq M(\leq m,n)\leq N(\leq m+1,n)$ for $m<0$ and $1\leq n\leq 100$. They also pointed out that these inequalities can be restated as the existence of a re-ordering $τ_n$ on the set of partitions of $n$ such that $|\text{crank}(λ)|-|\text{rank}(τ_n(λ))|=0$ or $1$ for all partitions $λ$ of $n$, that is, the rank and the crank are nearly equal distributions over partitions of $n$. In the study of the spt-function, Andrews, Dyson and Rhoades proposed a conjecture on the unimodality of the spt-crank, and they showed that this conjecture is equivalent to the inequality $N(\leq m,n)\leq M(\leq m,n)$ for $m<0$ and $n\geq 1$. We proved this conjecture by combiantorial arguments. In this paper, we prove the inequality $N(\leq m,n)\leq M(\leq m,n)$ for $m<0$ and $n\geq 1$. Furthermore, we define a re-ordering $τ_n$ of the partitions $λ$ of $n$ and show that this re-ordering $τ_n$ leads to the nearly equal distribution of the rank and the crank. Using the re-ordering $τ_n$, we give a new combinatorial interpretation of the function ospt$(n)$ defined by Andrews, Chan and Kim, which immediately leads to an upper bound for $ospt(n)$ due to Chan and Mao.
Recently, Garvan obtained two-variable Hecke-Rogers identities for three universal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these identities by using Bailey pair technology. In this paper, we give proofs of Garvan's identities by applying Bailey's transform with the conjugate Bailey pair of Warnaar and three Bailey pairs deduced from two special cases of $_6ψ_6$ given by Slater. In particular, we obtain a compact form of two-variable Hecke-Rogers identity related to $g_3(z;q)$, which imply the corresponding identity given by Garvan. We also extend these two-variable Hecke-Rogers identities into infinite families.
By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $η_{2k}(n)$ of ranks of partitions of $n$. Recently, Garvan introduced the $2k$-th symmetrized moment $μ_{2k}(n)$ of cranks of partitions of $n$ in the study of the higher-order spt-function $spt_k(n)$. In this paper, we give a combinatorial interpretation of $μ_{2k}(n)$. We introduce $k$-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that $μ_{2k}(n)$ equals the number of $(k+1)$-marked Dyson symbols of $n$. We then introduce the full crank of a $k$-marked Dyson symbol and show that there exist an infinite family of congruences for the full crank function of $k$-marked Dyson symbols which implies that for fixed prime $p\geq 5$ and positive integers $r$ and $k\leq (p-1)/2$, there exist infinitely many non-nested arithmetic progressions $An+B$ such that $μ_{2k}(An+B)\equiv 0\pmod{p^r}$.
By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $η_{2k}(n)$ of ranks of partitions of $n$ in terms of $(k+1)$-marked Durfee symbols of $n$. In this paper, we consider the $k$-th symmetrized positive moment $\barη_k(n)$ of ranks of partitions of $n$ which is defined as the truncated sum over positive ranks of partitions of $n$. As combintorial interpretations of $\barη_{2k}(n)$ and $\barη_{2k-1}(n)$, we show that for fixed $k$ and $i$ with $1\leq i\leq k+1$, $\barη_{2k-1}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being zero and $\barη_{2k}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being positive. The interpretations of $\barη_{2k-1}(n)$ and $\barη_{2k}(n)$ also imply the interpretation of $η_{2k}(n)$ given by Andrews since $η_{2k}(n)$ equals $\barη_{2k-1}(n)$ plus twice of $\barη_{2k}(n)$. Moreover, we obtain the generating functions of $\barη_{2k}(n)$ and $\barη_{2k-1}(n)$.
The spt-function $spt(n)$ was introduced by Andrews as the weighted counting of partitions of $n$ with respect to the number of occurrences of the smallest part. Andrews, Garvan and Liang defined the spt-crank of an $S$-partition which leads to combinatorial interpretations of the congruences of $spt(n)$ mod 5 and 7. Let $N_S(m,n)$ denote the net number of $S$-partitions of $n$ with spt-crank $m$. Andrews, Garvan and Liang showed that $N_S(m,n)$ is nonnegative for all integers $m$ and positive integers $n$, and they asked the question of finding a combinatorial interpretation of $N_S(m,n)$. In this paper, we introduce the structure of doubly marked partitions and define the spt-crank of a doubly marked partition. We show that $N_S(m,n)$ can be interpreted as the number of doubly marked partitions of $n$ with spt-crank $m$. Moreover, we establish a bijection between marked partitions of $n$ and doubly marked partitions of $n$. A marked partition is defined by Andrews, Dyson and Rhoades as a partition with exactly one of the smallest parts marked. They consider it a challenge to find a definition of the spt-crank of a marked partition so that the set of marked partitions of $5n+4$ and $7n+5$ can be divided into five and seven equinumerous classes. The definition of spt-crank for doubly marked partitions and the bijection between the marked partitions and doubly marked partitions leads to a solution to the problem of Andrews, Dyson and Rhoades.
The notion of the spt-crank of a vector partition, or an $S$-partition, was introduced by Andrews, Garvan and Liang. Let $N_S(m,n)$ denote the number of $S$-partitions of $n$ with spt-crank $m$. Andrews, Dyson and Rhoades conjectured that $\{N_S(m,n)\}_m$ is unimodal for any $n$, and they showed that this conjecture is equivalent to an inequality between the rank and the crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and the crank of ordinary partitions, which implies $N_S(m,n)\geq N_S(m+1,n)$ for sufficiently large $n$ and fixed $m$. In this paper, we introduce a representation of an ordinary partition, called the $m$-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades by considering two cases. For $m\geq 1$, we construct an injection from the set of ordinary partitions of $n$ such that $m$ appears in the rank-set to the set of ordinary partitions of $n$ with rank not less than $-m$. The case for $m=0$ requires five more injections. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.
Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of $n$ with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of PD(3n+2) which implies the congruence of Andrews, Lewis and Lovejoy. For PD(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of $(1-q^{2n+1})$ times a function $F(q^2)$. We find an explicit formula for $F(q^2)$, which leads to a formula for the generating function of PD(3n). We also obtain a formula for the generating function of PD(3n+1). Our proofs rely on Chan's identity on Ramanujan's cubic continued fraction and some identities on cubic theta functions. By introducing a rank for the partitions with designed summands, we give a combinatorial interpretation of the congruence of Andrews, Lewis and Lovejoy.
Stanley defined a partition function t(n) as the number of partitions $λ$ of n such that the number of odd parts of $λ$ is congruent to the number of odd parts of the conjugate partition $λ'$ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n)-t(n). As a consequence, we see that t(n) has the same parity as the ordinary partition function p(n) for any n. A simple combinatorial explanation of this fact is also provided.
We find two involutions on partitions that lead to partition identities for Ramanujan's third order mock theta functions $ϕ(-q)$ and $ψ(-q)$. We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third order mock theta functions are consequences of these partition identities. Our combinatorial constructions also apply to Andrews' generalizations of Ramanujan's identities.
We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.
A recursively palindromic (RP) word is one that is a palindrome and whose left half-word and right half-word are each RP. Thus ABACABA is, and MADAM is not, an RP word. We count RP words of given length over a finite alphabet and RP compositions of an integer. We use the same method to determine the parity of the Catalan numbers.