Related papers: Rook placements in Young diagrams and permutation …

We study asymptotics of random shifted Young diagrams which correspond to a given sequence of reducible projective representations of the symmetric groups. We show limit results (Law of Large Numbers and Central Limit Theorem) for their…

There are numerous combinatorial objects associated to a Grassmannian permutation $w_\lambda$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations…

For fixed positive integers $n,m$, let $\mathrm{Mat}_{n\times m}(\mathbb{C})$ be the affine space consisting of all $n\times m$ complex matrices, and let $\mathbb{C}[\mathbf{x}_{n\times m}]$ be its coordinate ring. For $0\le…

We investigate asymptotic probabilistic phenomena arising from the application of the Schensted row insertion algorithm, a key component of the Robinson-Schensted-Knuth (RSK) correspondence, to random inputs. Our analysis centers on a…

A placement of chess pieces on a chessboard is called dominating, if each free square of the chessboard is under attack by at least one piece. In this contribution we compute the number of dominating arrangements of $k$ rooks on an $n\times…

Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…

We introduce an object called a decorated Young tableau which can equivalently be viewed as a continuous time trajectory of Young diagrams or as a non-intersecting line ensemble. By a natural extension of the Robinson-Schensted…

This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence…

I describe an empirical-yet-rigorous, algorithm, based on Riordan's rook polynomials and the so-called C-finite ansatz, fully implemented in the accompanying Maple package (http://www.math.rutgers.edu/~zeilberg/tokhniot/MENAGES ), MENAGES,…

We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $\lambda$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference…

We consider the decomposition into irreducible components of the external power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young…

We reconsider in modern terms the old discovery by A. Kirillov and M. Noumi, who devised peculiar operators adding columns to Young diagrams enumerating the Schur, Jack and Macdonald polynomials. In this sense, these are a kind of…

Monads play an important role in both the syntax and semantics of modern functional programming languages. The problem of combining them has been of profound interest at least since the 90s, and different approaches have been employed to…

We consider a random model for directed graphs whereby an arc is placed from one vertex to another with a prescribed probability which may vary from arc to arc. Using perturbation bounds as well as Chernoff inequalities, we show that the…

We give an algorithmic-bijective proof of Macdonald's reduced word identity in the theory of Schubert polynomials, in the special case where the permutation is dominant. Our bijection uses a novel application of David Little's generalized…

We calculate the probability distributions for the number of occurrences $n$ of a given $l$ letter word in a random string of $k$ letters. Analytical expressions for the distribution are known for the asymptotic regimes (i) $k \gg r^l \gg…

Counting permutations of $[n]$ by the number of records, i.e. left-to-right maxima, is a classic problem in combinatorial enumeration. In the first volume of ``The Art of Computer Programming", Donald Knuth demonstrated its relevance for…

We extend Friedman's theorem to show that, for any fixed $r>1$, a random $2r$--regular Schreier graph associated with the action of $r$ uniformly random permutations of $[n]$ on $k_{n}$--tuples of distinct elements in $[n]$ has a…

We extend the results of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady, and Mori. We consider a model of random tree growth, where at each time unit a new node is added and attached to an already existing node chosen at random. The…

The study of permutation and partition statistics is a classical topic in enumerative combinatorics. The major index statistic on permutations was introduced a century ago by Percy MacMahon in his seminal works. In this extended abstract,…