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papers

Publications (42)

math.CO2022

Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions

Shane Chern, Shishuo Fu, Zhicong Lin

Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$ -avoiding in…

math.CO2025

Parity patterns meet Genocchi numbers, I: four labelings and three bijections

Quan Yuan, Qi Fang, Shishuo Fu +1

Hetyei introduced in 2019 the homogenized Linial arrangement and showed that its regions are counted by the median Genocchi numbers. In the course of devising a different proof of…

math.CO2020

Refined Wilf-equivalences by Comtet statistics

Shishuo Fu, Zhicong Lin, Yaling Wang

We launch a systematic study of the refined Wilf-equivalences by the statistics $comp$ and $iar$ , where $comp (I ¨ )$ and $iar (I ¨ )$ are the number…

math.CO2018

On certain unimodal sequences and strict partitions

Shishuo Fu, Dazhao Tang

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions wi…

math.CO2017

Multiranks and classical theta functions

Shishuo Fu, Dazhao Tang

Multiranks and new rank/crank analogs for a variety of partitions are given, so as to imply combinatorially some arithmetic properties enjoyed by these types of partitions. Our met…

math.CO2024

Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities

Shishuo Fu, Haijun Li

Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a doubl…

math.CO2021

A combinatorial bijection on di-sk trees

Shishuo Fu, Zhicong Lin, Yaling Wang

A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $⊖$ , and no node has the same label as its right child. The di-sk trees are in natural bijection w…

math.CO2019

On two unimodal descent polynomials

Shishuo Fu, Zhicong Lin, Jiang Zeng

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $I^^{3}$ -coefficients of the first are positive wit…

math.CO2017

On a generalized crank for $k$ -colored partitions

Shishuo Fu, Dazhao Tang

A generalized crank ( $k$ -crank) for $k$ -colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined $k$ -crank. N…

math.CO2017

Generalizing a partition theorem of Andrews

Shishuo Fu, Dazhao Tang

Motivated by Andrews' recent work related to Euler's partition theorem, we consider the set of partitions of an integer $n$ where the set of even parts has exactly $j$ elements, ve…

math.CO2016

Partitions with fixed largest hook length

Shishuo Fu, Dazhao Tang

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 siz…

math.CO2017

Mahonian STAT on rearrangement class of words

Shishuo Fu, Ting Hua, Vincent Vajnovszki

In 2000, Babson and SteingrÃmsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed…

math.CO2026

Signed counting of partition matrices

Shane Chern, Shishuo Fu

We prove that the signed counting (with respect to the parity of the `` $inv$ '' statistic) of partition matrices equals the cardinality of a subclass of inversion seq…

math.CO2024

A group action on cyclic compositions and $I^^{3}$ -positivity

Shishuo Fu, Jie Yang

Let $w_{n, k, m}$ be the number of Dyck paths of semilength $n$ with $k$ occurrences of $U D$ and $m$ occurrences of $U U D$ . We establish in two ways a new interpretation of the number…

math.CO2011

The negative q-binomial

Shishuo Fu, Victor Reiner, Dennis Stanton +1

Interpretations for the q-binomial coefficient evaluated at -q are discussed. A (q,t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary…

math.CO2026

Symmetric statistics on rational Dyck paths

Lilan Dai, Shishuo Fu, Dun Qiu

Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geo…

math.CO2017

Some inequalities for $k$ -colored partition functions

Shane Chern, Shishuo Fu, Dazhao Tang

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$ -colored partition functions $p_{- k} (n)$ for all $k \geq 2$ . This enables us to ext…

math.CO2025

Consecutive and quasi-consecutive patterns: $des$ -Wilf classifications and generating functions

Yan Wang, Qi Fang, Shishuo Fu +2

Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as e…

math.CO2019

Signed Mahonian polynomials for major and sorting indices

Huilan Chang, Sen-Peng Eu, Shishuo Fu +2

We derive some new signed Mahonian polynomials over the complex reflection group $G (r, 1, n) = C_{r} ≀ S_{n}$ , where the "sign" is taken to be any of the $2 r$ $1$ -dim characters…

math.CO2019

From $q$ -Stirling numbers to the Delta Conjecture: a viewpoint from vincular patterns

Joanna N. Chen, Shishuo Fu

The distribution of certain Mahonian statistic (called $BAST$ ) introduced by Babson and SteingrÃmsson over the set of permutations that avoid vincular pattern $1\underlin…

math.CO2026

On mesh patterns of short length: Equidistribution and enumeration

Qi Fang, Shishuo Fu, Sergey Kitaev +3

The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involutio…

math.CO2024

Sequences of odd length in strict partitions II: the $2$ -measure and refinements of Euler's theorem

Shishuo Fu, Haijun Li

The number of sequences of odd length in strict partitions (denoted as $sol$ ), which plays a pivotal role in the first paper of this series, is investigated in different c…

math.CO2025

A refined view of a curious identity for partitions into odd parts with designated summands

Shishuo Fu, James Sellers

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted intege…

math.CO2019

Bijective recurrences concerning two SchrÃ¶der triangles

Shishuo Fu, Yaling Wang

Let $r (n, k)$ (resp. $s (n, k)$ ) be the number of SchrÃ¶der paths (resp. little SchrÃ¶der paths) of length $2 n$ with $k$ hills, and set $r (0, 0) = s (0, 0) = 1$ . We bijectively establish the…

math.CO2017

A unifying combinatorial approach to refined little GÃ¶llnitz and Capparelli's companion identities

Shishuo Fu, Jiang Zeng

Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little GÃ¶llnitz identities. Noticing the conn…

math.CO2019

A lecture hall theorem for $m$ -falling partitions

Shishuo Fu, Dazhao Tang, Ae Ja Yee

For an integer $m \geq 2$ , a partition $I^» = (I^»_{1}, I^»_{2}, \dots)$ is called $m$ -falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $I^»_{i}$ 's form a nonin…

math.CO2026

Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities

Shishuo Fu, Chenwei Wang

Signed enumerations of l-regular partitions by the parity of the number of parts are known to correspond to partitions with congruence conditions. Inspired by the recent combinator…

math.NT2023

Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

Shishuo Fu, Dazhao Tang

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer p…

math.CO2025

On fourteen equidistribution conjectures of Lv and Zhang and monotone mesh patterns with corner shadings

Qi Fang, Shishuo Fu, Sergey Kitaev +1

Three complementation-like involutions are constructed on permutations to prove, and in some cases generalize, all remaining fourteen joint symmetric equidistribution conjectures o…

math.CO2018

Multi-dimensional $q$ -summations and multi-colored partitions

Shane Chern, Shishuo Fu, Dazhao Tang

Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contain…

math.CO2025

An involution for trivariate symmetries of vincular patterns

Joanna N. Chen, Shishuo Fu, Jiang Zeng

We provide a bijective proof of the equidistribution of two pairs of vincular patterns in permutations, thereby resolving a recent open problem of Bitonti, Deb, and Sokal (arXiv:24…

math.CO2018

$(q, t)$ -Catalan numbers: gamma expansions, pattern avoidance and the $(- 1)$ -phenomenon

Shishuo Fu, Dazhao Tang, Bin Han +1

The aim of this paper is two-fold. We first prove several new interpretations of a kind of $(q, t)$ -Catalan numbers along with their corresponding $I^^{3}$ -expansions using pattern avoi…

math.CO2019

A new decomposition of ascent sequences and Euler--Stirling statistics

Shishuo Fu, Emma Yu Jin, Zhicong Lin +2

As shown by Bousquet-MÃ©lou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $(2 + 2)$ -free posets. It is known that ascent sequences are enumerated by th…

math.CO2026

Convolutive sequences, II: Parametrizations

Shane Chern, Dennis Eichhorn, Shishuo Fu +1

In recent work, the authors defined a sequence $(a_{n})_{n \geq 0}$ to be $m$ -convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end…

math.CO2024

Permanent identities, combinatorial sequences, and permutation statistics

Shishuo Fu, Zhicong Lin, Zhi-Wei Sun

In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers.…

math.CO2022

Combinatorial proofs and refinements of three partition theorems of Andrews

Shishuo Fu

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 t…

math.CO2021

Rooted quasi-Stirling permutations of general multisets

Shishuo Fu, Yanlin Li

Given a general multiset $M = {1^{m_{1}}, 2^{m_{2}}, \dots, n^{m_{n}}}$ , where $i$ appears $m_{i}$ times, a multipermutation $I ¨ $ of $M$ is called {\em quasi-Stirling}, i…

math.CO2020

$k$ -arrangements, statistics and patterns

Shishuo Fu, Guo-Niu Han, Zhicong Lin

The $k$ -arrangements are permutations whose fixed points are $k$ -colored. We prove enumerative results related to statistics and patterns on $k$ -arrangements, confirming several co…

math.CO2025

Convolutive sequences, I: Through the lens of integer partition functions

Shane Chern, Dennis Eichhorn, Shishuo Fu +1

Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}…

math.CO2020

On $120$ -avoiding inversion and ascent sequences

Zhicong Lin, Shishuo Fu

Recently, Yan and the first named author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length $3$ , where two of the thr…

math.CO2026

When arrow patterns meet classical patterns

Shishuo Fu, Zhenghe Yang

Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known a…

math.CO2022

An involution on restricted Laguerre histories and its applications

Joanna N. Chen, Shishuo Fu

Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, a…

Publications (42)

Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions

Parity patterns meet Genocchi numbers, I: four labelings and three bijections

Refined Wilf-equivalences by Comtet statistics

On certain unimodal sequences and strict partitions

Multiranks and classical theta functions

Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities

A combinatorial bijection on di-sk trees

On two unimodal descent polynomials

On a generalized crank for k-colored partitions

Generalizing a partition theorem of Andrews

Partitions with fixed largest hook length

Mahonian STAT on rearrangement class of words

Signed counting of partition matrices

A group action on cyclic compositions and I^3-positivity

The negative q-binomial

Symmetric statistics on rational Dyck paths

Some inequalities for k-colored partition functions

Consecutive and quasi-consecutive patterns: des-Wilf classifications and generating functions

Signed Mahonian polynomials for major and sorting indices

From q-Stirling numbers to the Delta Conjecture: a viewpoint from vincular patterns

On mesh patterns of short length: Equidistribution and enumeration

Sequences of odd length in strict partitions II: the 2-measure and refinements of Euler's theorem

A refined view of a curious identity for partitions into odd parts with designated summands

Bijective recurrences concerning two SchrÃ¶der triangles

A unifying combinatorial approach to refined little GÃ¶llnitz and Capparelli's companion identities

A lecture hall theorem for m-falling partitions

Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities

Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

On fourteen equidistribution conjectures of Lv and Zhang and monotone mesh patterns with corner shadings

Multi-dimensional q-summations and multi-colored partitions

An involution for trivariate symmetries of vincular patterns

(q,t)-Catalan numbers: gamma expansions, pattern avoidance and the (−1)-phenomenon

A new decomposition of ascent sequences and Euler--Stirling statistics

Convolutive sequences, II: Parametrizations

Permanent identities, combinatorial sequences, and permutation statistics

Combinatorial proofs and refinements of three partition theorems of Andrews

Rooted quasi-Stirling permutations of general multisets

k-arrangements, statistics and patterns

Convolutive sequences, I: Through the lens of integer partition functions

On 12​0-avoiding inversion and ascent sequences

When arrow patterns meet classical patterns

An involution on restricted Laguerre histories and its applications

On a generalized crank for $k$ -colored partitions

A group action on cyclic compositions and $I^^{3}$ -positivity

Some inequalities for $k$ -colored partition functions

Consecutive and quasi-consecutive patterns: $des$ -Wilf classifications and generating functions

From $q$ -Stirling numbers to the Delta Conjecture: a viewpoint from vincular patterns

Sequences of odd length in strict partitions II: the $2$ -measure and refinements of Euler's theorem

A lecture hall theorem for $m$ -falling partitions

Multi-dimensional $q$ -summations and multi-colored partitions

$(q, t)$ -Catalan numbers: gamma expansions, pattern avoidance and the $(- 1)$ -phenomenon

$k$ -arrangements, statistics and patterns

On $120$ -avoiding inversion and ascent sequences