Related papers: Rank partition functions and truncated theta ident…
Denote by $p(n)$ the number of partitions of $n$ and by $N(a,M;n)$ the number of partitions of $n$ with rank congruent to $a$ modulo $M$. We find and prove a general formula for Dyson's ranks by considering the deviation of the ranks from…
A permutation of length $n$ is called a flattened partition if the leading terms of maximal chains of ascents (called runs) are in increasing order. We analogously define flattened parking functions: a subset of parking functions for which…
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive…
Euler's classic partition identity states that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We develop a new generalization of this identity, which yields a previous…
For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After…
A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of…
We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…
Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well…
The rank of a skew partition $\lambda/\mu$, denoted $rank(\lambda/\mu)$, is the smallest number $r$ such that $\lambda/\mu$ is a disjoint union of $r$ border strips. Let $s_{\lambda/\mu}(1^t)$ denote the skew Schur function…
Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers-Ramanujan-type identities, we obtain the $\mathrm{A}_2$ (or $\mathrm{A}_2^{(1)}$) analogues of the celebrated Andrews-Gordon…
Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…
We consider the problem of learning an unknown partition of an $n$ element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple…
Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and Sellers. Here, we…
In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of Euclidean space is the restriction of a function that is continuously differentiable to order p. A necessary and sufficient…
The counting of partitions according to their genus is revisited. The case of genus 0 -- non-crossing partitions -- is well known. Our approach relies on two pillars: first a functional equation between generating functions, originally…
Integer partitions have long been of interest to number theorists, perhaps most notably Ramanujan, and are related to many areas of mathematics including combinatorics, modular forms, representation theory, analysis, and mathematical…
We find the number of partitions of $n$ whose BG-rank is $j$, in terms of $pp(n)$, the number of pairs of partitions whose total number of cells is $n$, giving both bijective and generating function proofs. Next we find congruences mod 5…
It is proved that the number of partitions of n with odd mex and k parts that aren't ones equals the number of partitions of n with nonnegative crank and k parts that aren't ones..
In this paper, we interpret the theta rank of an irreducible character of a finite classical group in terms of the data from the Lusztig classification. Then we prove the following two results: (1) the agreement of the $U$-rank and the…
We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of…