Related papers: Hook-length formula and applications to alternatin…

Finding diverse solutions in combinatorial problems recently has received considerable attention (Baste et al. 2020; Fomin et al. 2020; Hanaka et al. 2021). In this paper we study the following type of problems: given an integer $k$, the…

We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics,…

Some features of extended loops are considered. In particular, the behaviour under diffeomorphism transformations of the wavefunctions with support on the extended loop space are studied. The basis of a method to obtain analytical…

The alternating-runs polynomial enumerates alternating runs in the symmetric group. There are three formulae for the number of permutations, $R_{n,k}$ in $\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. We show…

Using only a symmetric p-core partition and p-quotient, we give an explicit formula for the set of diagonal hook lengths of the associated symmetric partition.

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: \frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}(h_u^2 - i^2) = \frac{1}{2(r+1)^2}…

We introduce the classes of descendingly flexible and descendingly alternative algebras over an arbitrary field $\mathbb{F}$. We suggest a new method based on the sequence of differences between the dimensions of the linear spans of words,…

We deduce a special case of a theorem of M. Haiman concerning alternating polynomials in 2n variables from our results about almost commuting variety, obtained earlier in a joint work with W.-L. Gan.

We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number $n$ of elements, or a…

We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.

The classical hook length formula of enumerative combinatorics expresses the number of standard Young tableaux of a given partition shape as a single fraction. In recent years, two generalizations of this formula have emerged: one by Pak…

This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where…

In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family…

We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work…

Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the…

Call a permutation $k$-inflatable if the sequence of its tensor products with uniform random permutations of increasing lengths has uniform $k$-point pattern densities. Previous work has shown that nontrivial $k$-inflatable permutations do…

We consider a novel version of the classical Jacquet-Langlands {correspondence}, explore a $p$-adic extension of the Jacquet-Langlands correspondence, and as an explicit example we find an overconvergent automorphic form of weight~1 which…

The paper begins with a novel variational formulation of Duffing equation using the extended framework of Hamilton's principle (EHP). This formulation properly accounts for initial conditions, and it recovers all the governing differential…

Descents in permutations or words are defined from the relative position of two consecutive letters. We investigate a statistic involving patterns of k consecutive letters, and show that it leads to Hopf algebras generalizing noncommutative…

X.S. Lin and O. Dasbach proved that the sum of the absolute value of the second and penultimate coefficients of the Jones polynomial of an alternating knot is equal to the twist number of the knot. In this paper we give a new proof of their…