Ian Goulden, Alexander Yong
We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture, by means of a construction involving Dyck paths, a particular type of lattice path.
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An $n$-multiset of $[k]=\{1,2,\ldots, k\}$ consists of a set of $n$ elements from $[k]$ where each element can be repeated. We present the bivariate generating function for $n$-multisets of $[k]$ with no consecutive elements. For $n=k$, these multisets have the same enumeration as directed animals in the square lattice. Then we give constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern $DUD$, and we show how classical and novel statistics are transported by these bijections.
It is well-known that two conjugate partitions have the same hook multiset. But two different partitions with the same hook multiset may not be conjugate to each other. In $1977$, Herman and Chung proposed the following question: What are the necessary and sufficient conditions for partitions to be determined by their hook multisets up to conjugation? In this paper, we will answer this question by giving a criterion to determine whether two different partitions with the same hook multiset are conjugate to each other.
A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal, Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset hook-content formula is also proved.
Valid hook configurations are combinatorial objects used to understand West's stack sorting map as well as cumulants in noncommutative probability theory. We show a bijection between reduced valid hook configurations on 312-avoiding permutations with the maximal allowed number of points and 3D-Dyck words, proving a conjecture of Sankar's. We extend to a bijection between all 312-avoiding reduced valid hook configurations and 3D-Dyck words with specified modifications. We show how these can be counted in terms of the number of 3D-Dyck words of length 3k in which exactly i Y's do not have an X i…
Motivated by the question of finding a type B analogue of the bijection between oscillating tableaux and matchings, we find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m-rim hook tableau is defined as a sequence $(λ^0,λ^1,...,λ^{2n})$ of Young diagrams starting with the empty shape and ending with the empty shape such that $λ^{i}$ is obtained from $λ^{i-1}$ by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook ta…
We calculate the total number of humps in Dyck and in Motzkin paths, and we give Standard-Young-Tableaux-interpretations of the numbers involved. One then observes the intriguing phenomena that the humps-calculations change the partitions in a strip to partitions in a hook.
Short, bijective proofs of identities for multisets of `hook pairs' (arm-leg pairs) of the cells of certain diagrams are given. These hook pair identities were originally found by Regev.
In this paper, we take interest in finding applications for a hook-length formula recently proved in (Morales Pak Panova 2016). This formula can be applied to give a non trivial relation between alternating permutations and weighted Dyck paths. First, we give an alternative proof for this result using continued fractions, and then we apply a similar reasoning to the more general case of $k$-alternating permutations.
In this short note, we give large counterexamples to natural questions about certain order polytopes, in particular, Gelfand--Tsetlin polytopes. Several of the counterexamples are too large to be discovered via a brute-force computer search. We also show that the multiset of hooks in a Young diagram is not enough information to determine the Ehrhart polynomial for an associated order polytope. This is somewhat counter-intuitive to the fact that the multiset of hooks always determine the leading coefficient of the Ehrhart polynomial.
Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal; $(m,n)$-cores are partitions with no hook length equal to $m$ or $n$.Anderson established a bijection between $(m,n)$-Dyck paths and $(m,n)$-cores. We define a new transformation, called rank complement, on rational Dyck paths. We show that rank complement corresponds to conjugation of $(m,n)$-cores under Anderson's bijection. This leads to: i) a …
Introduced by Ardila (J. Combin. Theory Ser. A, 2003), the Catalan matroid is obtained by defining the bases of the matroid using Dyck paths from $(0,0)$ to $(n,n)$. Further research has gone into the topic, with variants like lattice path matroids (introduced by Bonin, de Mier, and Noy (J. Combin. Theory Ser. A, 2003)) and shifted matroids (introduced independently by Klivans (2003), and Ardila) being studied intensively. In this short note, we introduce the Young matroid, an extension of the Catalan matroid, where the bases are defined using the standard Young tableaux of a fixed shape. This…
The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations.
For non-negative integers $n$ and $k$ with $n \ge k$, a {\em $k$-minor} of a partition $λ= [λ_1, λ_2, \dots]$ of $n$ is a partition $μ= [μ_1, μ_2, \dots]$ of $n-k$ such that $μ_i \le λ_i$ for all $i$. The multiset $\widehat{M}_k(λ)$ of $k$-minors of $λ$ is defined as the multiset of $k$-minors $μ$ with multiplicity of $μ$ equal to the number of standard Young tableaux of skew shape $λ/ μ$. We show that there exists a function $G(n)$ such that the partitions of $n$ can be reconstructed from their multisets of $k$-minors if and only if $k \le G(n)$. Furthermore, we prove that $\lim_{n \rightarro…
How to bounce your canon permutation
本文研究了一类新的回文下降多项式，通过Dyck路径和排列生成因果排列，并探讨了其多项式性质及组合解释。
$λ$-shaped random matrices, $λ$-plane trees, and $λ$-Dyck paths
本文研究了$λ$-形随机矩阵的分布特性，揭示了$λ$-平面树与$λ$-Dyck路径的双射关系，并在肥钩形状情况下给出了分布的显式公式。
Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer $n$ has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number $t$ is absent from the diagram then the partition is called a $t$-core. A partition is an $(s,t)$-core if it is both an $s$- and a $t$-core. Since the work of Anderson on $(s,t)$-cores, the topic has received a growing attention. This paper expands the discussion to multi…
Inequalities among two rowed immanants of the $q$-Laplacian of Trees and Odd height peaks in generalized Dyck paths
本文证明了树$q$-拉普拉斯算子归一化immanant在两行划分下的不等式序列，主要引理涉及二项式系数与不可约特征标，并给出了Riordan路径及广义Dyck路径奇数高度峰的组合与概率解释。
On the area-depth symmetry on Łukasiewicz paths
通过Łukasiewicz路径与平面树之间的经典双射，证明了固定第一步和最后一步的Łukasiewicz路径上area和depth统计量具有联合等分布性，从而解决了Qu和Zhang（2025）提出的猜想。
We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also g…
In this manuscript we consider the set of Dyck paths equipped with the uniform measure, and we study the statistical properties of a deformation of the observable "area below the Dyck path" as the size $N$ of the path goes to infinity. The deformation under analysis is apparently new: while usually the area is constructed as the sum of the heights of the steps of the Dyck path, here we regard it as the sum of the lengths of the connected horizontal slices under the path, and we deform it by applying to the lengths of the slices a positive regular function $ω(\ell)$ such that $ω(\ell) \sim \ell…
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