arXiv subjects

Zhicong Lin

Publications and source records attributed to Zhicong Lin.

At least 19 recordsLinked to original sources

arXiv2026

Bijective proofs of several conjectures on Jacobi permutations

Jacobi permutations, invented by Viennot in the context of the Jacobi elliptic functions, are counted by the Euler numbers. Recently, Henke, Hoffman, Stephens, Yuan, and Zhuang studied refined enumerations of Jacobi permutations and proposed three conjectures concerning the distribution of several statistics on Jacobi permutations. In this paper, we prove these conjectures by establishing explicit bijections involving increasing even trees, increasing binary trees, alternating permutations, and Andr\'e permutations. One highlight of our results is a bijection between Jacobi permutations and Andr\'e I permutations that transforms the pair of statistics $(\Ascbot, \last)$ to the pair of statistics $(\Desbot, \first)$. Here the statistic $\Ascbot$ (resp., $\Desbot$) denotes the set of ascent bottoms (resp., descent bottoms) of permutations, and the statistic $\first$ (resp., $\last$) denotes the first (resp., $\last$) letter of permutations. Furthermore, we investigate pairs of statistics on Andr\'e permutations and simsun permutations that are equidistributed with the pair $(\asc, \last)$ on Jacobi permutations, where $\asc$ denotes the number of ascents of permutations. Finally, we obtain a closed-form formula for the trivariate exponential generating function of Jacobi permutations with respect to the number of ascents and the numbers of letters smaller and larger than the last letter.

math.CO↗

arXiv2026

Proof of a Conjecture on Young Tableaux with Walls

Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this work, we prove a conjecture of Fuchs and Yu concerning the enumeration of two classes of three-row Young tableaux with walls. Together with the work of Chang, Fuchs, Liu, Wallner, and Yu, our result verifies a conjecture of Pons and Batle on tree-child networks. This conjecture had been regarded as a specific and challenging problem in the phylogenetics community until its resolution in the present work.

math.CO↗

arXiv2025

An involution for a Catalan-tangent number identity

We provide an involution proof of a Catalan-tangent number identity arising from the study of peak algebra that was found by Aliniaeifard and Li. In the course, we find a new combinatorial identity for the tangent numbers $T_{2n+1}$: $$ \sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}. $$ Moreover, we derive two different $q$-analogs of the above identity from the combinatorial perspective.

math.CO↗

arXiv2025

An identity relating Catalan numbers to tangent numbers with arithmetic applications

We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity $$ \sum_{k=0}^{n-1}{2n\choose 2k+1}2^{2n-2k}(-1)^{k}E_{2k+1}=2^{2n+1}, $$ where $E_{2k+1}$ denote the tangent numbers. Interestingly, the latter identity can be applied to prove that $(n + 1)E_{2n+1}$ is divisible by $2^{2n}$ and the quotient is an odd number, a fact whose traditional proofs require significant calculations. Moreover, we find a natural $q$-analog of the latter identity with a combinatorial proof. This $q$-identity can be applied to prove Foata's divisibility property of the $q$-tangent numbers, which responds to a problem raised by Sch\"utzenberger.

math.CO↗

arXiv2025

A bijection between $321$- and $213$-avoiding permutations preserving $t$-stack-sortability

We construct a bijection between $321$- and $213$-avoiding permutations that preserves the property of $t$-stack-sortability. Our bijection transforms natural statistics between these two classes of permutations and proves a refinement of an enumerative conjecture posed by Zhang and Kitaev. This work contributes further to the long-standing line of research on bijections between length-3 pattern avoiding permutations. Increasing binary trees lie at the heart of our approach.

math.CO↗

arXiv2025

Bijections in weakly increasing trees via binary trees

As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin-Ma-Ma-Zhou in 2021. Motived by some symmetries in plane trees proved recently by Dong, Du, Ji and Zhang, we construct four bijections on weakly increasing trees in the same flavor via switching the role of left child and right child of some specified nodes in their corresponding binary trees. Consequently, bijective proofs of the aforementioned symmetries found by Dong et al. and a non-recursive construction of a bijection on plane trees of Deutsch are provided. Applications of some symmetries in weakly increasing trees to permutation patterns and statistics will also be discussed.

math.CO↗

arXiv2025

Web permutations, Seidel triangle and normalized $\gamma$-coefficients

The web permutations were introduced by Hwang, Jang and Oh to interpret the entries of the transition matrix between the Specht and $\mathrm{SL}_2$-web bases of the irreducible $\S_{2n}$-representation indexed by $(n,n)$. They conjectured that certain classes of web permutations are enumerated by the Seidel triangle. Using generating functions, Xu and Zeng showed that enumerating web permutations by the number of drops, fixed points and cycles gives rise to the normalized $\gamma$-coefficients of the $(\alpha,t)$-Eulerian polynomials. They posed the problems to prove their result combinatorially and to find an interpretation of the normalized $\gamma$-coefficients in terms of cycle-up-down permutations. In this work, we prove the enumerative conjecture of Hwang-Jang-Oh and answer the two open problems proposed by Xu and Zeng.

math.CO↗

arXiv2025

Combinatorics on bi-$\gamma$-positivity of $1/k$-Eulerian polynomials

The $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ were introduced as ascent polynomials over $k$-inversion sequences by Savage and Viswanathan. The bi-$\gamma$-positivity of the $1/k$-Eulerian polynomials $A^{(k)}_{n}(x)$ was known but to give a combinatorial interpretation of the corresponding bi-$\gamma$-coefficients still remains open. The study of the theme of bi-$\gamma$-positivities from purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-$\gamma$-coefficients of $A^{(k)}_{n}(x)$ by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps: (i) construct a bijection between $k$-Stirling permutations and certain forests that are named increasing pruned even $k$-ary forests; (ii) introduce a generalized Foata--Strehl action on increasing pruned even $k$-ary trees which implies the longest ascent-plateau polynomials over $k$-Stirling permutations with initial letter $1$ are $\gamma$-positive, a result that may have independent interest; (iii) develop two crucial transformations on increasing pruned even $k$-ary forests to conclude our combinatorial interpretation.

math.CO↗

arXiv2024

Bijections around Springer numbers

Arnol'd proved in 1992 that Springer numbers enumerate the Snakes, which are type $B$ analogs of alternating permutations. Chen, Fan and Jia in 2011 introduced the labeled ballot paths and established a ``hard'' bijection with snakes. Callan conjectured in 2012 and Han--Kitaev--Zhang proved recently that rc-invariant alternating permutations are counted by Springer numbers. Very recently, Chen--Fang--Kitaev--Zhang investigated multi-dimensional permutations and proved that weakly increasing $3$-dimensional permutations are also counted by Springer numbers. In this work, we construct a sequence of ``natural'' bijections linking the above four combinatorial objects.

math.CO↗

arXiv2024

Parity statistics on restricted permutations and the Catalan--Schett polynomials

Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over $321$-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on $321$-avoiding permutations is solved and several combinatorial interpretations for the Catalan--Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.

math.CO↗

arXiv2024

On a conjecture concerning the $r$-Euler-Mahonian statistic on permutations

A pair $(\mathrm{st_1}, \mathrm{st_2})$ of permutation statistics is said to be $r$-Euler-Mahonian if $(\mathrm{st_1}, \mathrm{st_2})$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$, where $\mathrm{rdes}$ denotes the $r$-descent number and $\mathrm{rmaj}$ denotes the $r$-major index introduced by Rawlings. The main objective of this paper is to prove that $(\mathrm{exc}_r, \mathrm{den}_r)$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over $\mathfrak{S}_{n}$, thereby confirming a recent conjecture posed by Liu. When $r=1$, the result recovers the equidistribution of $(\mathrm{des}, \mathrm{maj})$ and $(\mathrm{exc}, \mathrm{den})$, which was first conjectured by Denert and proved by Foata and Zeilberger.

math.CO↗

arXiv2024

Constrained coding upper bounds via Goulden-Jackson cluster theorem

Motivated by applications in DNA-based data storage, constrained codes have attracted a considerable amount of attention from both academia and industry. We study the maximum cardinality of constrained codes for which the constraints can be characterized by a set of forbidden substrings, where by a substring we mean some consecutive coordinates in a string. For finite-type constrained codes (for which the set of forbidden substrings is finite), one can compute their capacity (code rate) by the ``spectral method'', i.e., by applying the Perron-Frobenious theorem to the de Brujin graph defined by the code. However, there was no systematic method to compute the exact cardinality of these codes. We show that there is a surprisingly powerful method arising from enumerative combinatorics, which is based on the Goulden-Jackson cluster theorem (previously not known to the coding community), that can be used to compute not only the capacity, but also the exact formula for the cardinality of these codes, for each fixed code length. Moreover, this can be done by solving a system of linear equations of size equal to the number of constraints. We also show that the spectral method and the cluster method are inherently related by establishing a direct connection between the spectral radius of the de Brujin graph used in the first method and the convergence radius of the generating function used in the second method. Lastly, to demonstrate the flexibility of the new method, we use it to give an explicit upper bound on the maximum cardinality of variable-length non-overlapping codes, which are a class of constrained codes defined by an infinite number of forbidden substrings.

cs.IT↗

arXiv2024

A $q$-analog of the Stirling-Eulerian Polynomials

In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial $A_n(x,y|\alpha,\beta)$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of $A_n(x,y|\alpha,\beta)$, denoted $P_n(u_1,u_2,u_3,u_4|\alpha,\beta)$, 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 $P_n(u_1,u_2,u_3,u_4|\alpha,\beta)$, generalizing the work of Carlitz and Scoville. Ji's formula has many nice consequences, one of which is an intriguing $\gamma$-positivity expansion for $A_n(x,y|\alpha,\beta)$. 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 $\gamma$-positivity expansion of $A_n(x,y|\alpha,\beta)$.

math.CO↗

arXiv2024

CLIP-driven Outliers Synthesis for few-shot OOD detection

Few-shot OOD detection focuses on recognizing out-of-distribution (OOD) images that belong to classes unseen during training, with the use of only a small number of labeled in-distribution (ID) images. Up to now, a mainstream strategy is based on large-scale vision-language models, such as CLIP. However, these methods overlook a crucial issue: the lack of reliable OOD supervision information, which can lead to biased boundaries between in-distribution (ID) and OOD. To tackle this problem, we propose CLIP-driven Outliers Synthesis~(CLIP-OS). Firstly, CLIP-OS enhances patch-level features' perception by newly proposed patch uniform convolution, and adaptively obtains the proportion of ID-relevant information by employing CLIP-surgery-discrepancy, thus achieving separation between ID-relevant and ID-irrelevant. Next, CLIP-OS synthesizes reliable OOD data by mixing up ID-relevant features from different classes to provide OOD supervision information. Afterward, CLIP-OS leverages synthetic OOD samples by unknown-aware prompt learning to enhance the separability of ID and OOD. Extensive experiments across multiple benchmarks demonstrate that CLIP-OS achieves superior few-shot OOD detection capability.

cs.CV↗

arXiv2023

The Binomial-Stirling-Eulerian Polynomials

We introduce the binomial-Stirling-Eulerian polynomials, denoted $\tilde{A}_n(x,y|{\alpha})$, which encompass binomial coefficients, Eulerian numbers and two Stirling statistics: the left-to-right minima and the right-to-left minima. When $\alpha=1$, these polynomials reduce to the binomial-Eulerian polynomials $\tilde{A}_n(x,y)$, originally named by Shareshian and Wachs and explored by Chung-Graham-Knuth and Postnikov-Reiner-Williams. We investigate the $\gamma$-positivity of $\tilde{A}_n(x,y|{\alpha})$ from two aspects: firstly by employing the grammatical calculus introduced by Chen; and secondly by constructing a new group action on permutations. These results extend the symmetric Eulerian identity found by Chung, Graham and Knuth, and the $\gamma$-positivity of $\tilde{A}_n(x,y)$ first demonstrated by Postnikov, Reiner and Williams.

math.CO↗

arXiv2023

Two involutions on binary trees and generalizations

This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection $\varphi$ between binary trees and plane trees answers an open problem posed by Bai and Chen. This involution can be generalized to weakly increasing trees, which admits to merge two recent equidistributions found by Bai--Chen and Chen--Fu, respectively. The other one is constructed to answer a bijective problem on di-sk trees asked by Fu--Lin--Wang and can be generalized naturally to rooted labeled trees. This second involution combined with $\varphi$ leads to a new statistic on plane trees whose distribution gives the Catalan's triangle. Moreover, a quadruple equidistribution on plane trees involving this new statistic is proved via a recursive bijection.

math.CO↗

arXiv2022

A bijection for length-$5$ patterns in permutations

A bijection between $(31245,32145,31254,32154)$-avoiding permutations and $(31425,32415,31524,32514)$-avoiding permutations is constructed, which preserves five classical set-valued statistics. Combining with two codings of permutations due respectively to Baril--Vajnovszki and Martinez--Savage proves an enumerative conjecture posed by Gao and Kitaev. Moreover, the generating function for the common counting sequence is proved to be algebraic.

math.CO↗

arXiv2022

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

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 inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case, and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.

math.CO↗