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Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, in the form of a continued fraction, for a fourteen-parameter family of such sequences and interpret these in…
We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1) and t_1...t_{m-2}(m-1)m in a permutation, respectively. By a simple involution in…
Given an edge-colored graph, the goal of the proportional fair matching problem is to find a maximum weight matching while ensuring proportional representation (with respect to the number of edges) of each color. The colors may correspond…
Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations. Hoffman established a link between Springer numbers,…
This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations selected…
In this paper we study the inverse of so-called unfair permutations, and explore various properties of them. Our investigation begins with comparing this class of permutations with uniformly random permutations, and showing that they behave…
We introduce and study deferred N\"{o}rlund statistical convergence in probability, mean of order $r,$ distribution, and study the interrelation among them. Based upon the proposed method to illustrate the findings, we present new Korovkin…
Hajnal and Szemer\'{e}di proved that if $G$ is a finite graph with maximum degree $\Delta$, then for every integer $k \geqslant \Delta+1$, $G$ has a proper coloring with $k$ colors in which every two color classes differ in size at most by…
We prove a formula expressing the Kerov polynomial $\Sigma_k$ as a weighted sum over the lattice of noncrossing partitions of the set $\{1,...,k+1\}$. In particular, such a formula is related to a partial order $\mirr$ on the Lehner's…
A $k$-Stirling permutation of order $n$ is said to be "flattened" if the leading terms of its increasing runs are in ascending order. We show that flattened $k$-Stirling permutations of order $n+1$ are in bijection correspondence with a…
We study cycle counts in permutations of $1,\dots,n$ drawn at random according to the Mallows distribution. Under this distribution, each permutation $\pi \in S_n$ is selected with probability proportional to $q^{\text{inv}(\pi)}$, where…
We have recently proposed arXiv:2105.11565 a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with $SU(N)$ gauge group. In this paper, we apply the developed…
We show that there is a permutation $f$ of the positive integers such that for $n \geq 2,$ l.c.m.$(f(n), f(n+1)) \leq cn(\log n)^2,$ where $c$ is a positive constant. It improves previous results of Erd\"os, Freud and Hegyvari (1983), and…
Finding distributions of statistics in pattern-avoiding permutations has attracted significant attention in the literature. In particular, Chen, Kitaev, and Zhang derived functional equations for the joint distributions of any subset of…
In this paper, we examine the unimodality and strict unimodality of certain formal bivariate Laurent series with non-negative coefficients. We show that the sets of these formal bivariate Laurent series form commutative semirings under the…
In this thesis, we apply the stack sorting operator to $r$-permutations and construct the functional equation for the generating function of two-stack-sortable $k$-tuple $r$-permutations counted by descents by using a factorization similar…
Simsun permutations, Andr\'e I permutations and Andr\'e II permutations are three combinatorial models for Euler numbers. It's known that the descent statistic is equidistributed over the set of Andr\'e I permutations and the set of simsun…
We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right…
Recently Albert and Bousquet-M\'elou \cite{AB15} obtained the solution to the long-standing problem of the number of permutations sortable by two stacks in parallel (tsip). Their solution was expressed in terms of functional equations. We…
The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson…