We consider $β$--Plancherel measures \cite{Ba.Ra.} on subsets of partitions -- and their asymptotics. These subsets are the Young diagrams contained in a $(k,\ell)$--hook, and we calculate the asymptotics of the expected shape of these diagrams, relative to such measures. We also calculate the asymptotics of the distribution function of the lengths of the rows and the columns for these diagrams. This might be considered as the restriction to the $(k,\ell)$--hook of the fundamental work of Baik, Deift and Johansson \cite{B.D.J.1}. The above asymptotics are given here by ratios of certain Selberg-type multi--integrals.
We construct a hook-content formula and its $q$-analog using excited Young diagrams analogous to Naruse's hook-length formula for skew shapes. Furthermore, we show that our hook-content formula has a simple factorization and give some conjectures and questions related to its $q$-analog.
Closed formulas are known for $S(k,0;n)$, the number of standard Young tableaux of size $n$ and with at most $k$ parts, where $1\le k\le 5$. Here we study the analogue problem for $S(k,\ell;n)$, the number of standard Young tableaux of size $n$ which are contained in the $(k,\ell)$ hook. We deduce some formulas for the cases $k+\ell\le 4$.
We introduce a new impartial game, named Multiple Hook Removing Game (MHRG for short). We also determine the $\mathcal{G}$-values of some game positions (including the starting positions) in MHRG$(m,n)$, the MHRG whose starting position is the rectangular Young diagram of size $m\times n$ with the unimodal numbering. In addition, we prove that MHRG$(m,n)$ is isomorphic, as games, to MHRG$(m,n+1)$ (if $m\le n$ and $m+n$ is even), and give a relationship between MHRG$(n,n+1)$ (and MHRG$(n,n)$) and HRG$(S_n)$, the Hook Removing Game in terms of shifted Young diagrams.
The asymptotics of the "$(k,\ell)$ hook" sums $S_{k,\ell}^{(2z)}(n)$ (see~\eqref{the.sums.1}) were calculated in~\cite{berele}. It was recently realized that in~\cite[Section 7]{berele} there are few misprints and certain confusion with regard to the notations, so that the precise asymptotics of $S_{k,\ell}^{(2z)}(n)$ is not clear. Here we add more details and carefully repeat these calculations, which lead to explicit values for the asymptotics of $S_{k,\ell}^{(2z)}(n).$
We present a conjectual hook formula concerning the number of the standard tableaux on "cylindric" skew diagrams. Our formula can be seen as an extension of Naruse's hook formula for skew diagrams. Moreover, we prove our conjecture in some special cases.
Let $p(n)$ be the number of all integer partitions of the positive integer $n$ and let $λ$ be a partition, selected uniformly at random from among all such $p(n)$ partitions. It is known that each partition $λ$ has a unique graphical representation, composed by $n$ non-overlapping cells in the plane called Young diagram. As a second step of our sampling experiment, we select a cell $c$ uniformly at random from among all $n$ cells of the Young diagram of the partition $λ$. For large $n$, we study the asymptotic behavior of the hook length $Z_n=Z_n(λ,c)$ of the cell $c$ of a random partituion $λ…
Multiple Hook Removing Game (MHRG for short) is an impartial game played in terms of Young diagrams. In this paper, we give a characterization of the set of all game positions in MHRG. As an application, we prove that for $t \in \mathbb{Z}_{\geq 0}$ and $m, n \in \mathbb{N}$ such that $t \leq m \leq n$, and a Young diagram $Y$ contained in the rectangular Young diagram $Y_{t,n}$ of size $t \times n$, $Y$ is a game position in MHRG with $Y_{m,n}$ the starting position if and only if $Y$ is a game position in MHRG with $Y_{t,n-m+t}$ the starting position, and also that the Grundy value of $Y$ in…
We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: \frac{1}{n!} \sum_{λ\vdash n} f_λ^2 \sum_{u \in λ} \prod_{i=1}^{r}(h_u^2 - i^2) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{r+1} \prod_{j=0}^{r} (n-j), where $f_λ$ is the number of standard Young tableaux of shape $λ$ and $h_u$ is the hook length of the square $u$ of the Young diagram of $λ$. We also obtain other similar formulas.
In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.
This is an etude in experimental semi-rigorous (rigorizable!) mathematics. The leading asymptotics was brilliantly derived by Allan Berele and Amitai Regev for general hooks H(k,l) and general powers z, but what about more refined asymptotics? For small k and l, one can "guess" a linear recurrence (since we live in the holonomic ansatz) and using the Birkhoff-Trjitzinsky method, beautifully implemented in Doron Zeilberger's Maple package AsyRec (that has been incorporated into the present Maple package), we computed amazing refined asymptotics, that confirm, with a vengeance, the Berele-Regev …
In this paper, we present a direct bijective proof of the hook-length formula for standard immaculate tableaux, which arose in the study of non-commutative symmetric functions. Our proof is along the spirit of Novelli, Pak and Stoyanovskii's combinatorial proof of the hook-length formula for standard Young tableaux.
There are $k$ kinds of length $k$ hooks with different arm length. Actually, this $k$ kinds appear uniformly in Young diagrams of size $n$. The property ``appear uniformly'' is called super symmetric. We give a combinatorial proof of the supper symmetric property of hook length.
A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called \emph{excited diagrams}, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. Recently, the formula for skew straight shapes was proved via a simple bumping algorithm. The aim of this paper is to extend this result to skew shifted shapes. Since straight skew shapes are special cases of skew shifted shapes, this is …
Recently, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. In this paper, we present a simple bijection that proves an equivalent recursive version of Naruse's result, in the same way that the celebrated hook-walk proof due to Green, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary sh…
We prove a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials. To do so, we prove an algebraic inequality, which we refer to as the "Slice and Push Inequality." This inequality compares expressions that come from Naruse's hook-length formula for the number of standard Young tableaux of a skew shape.
We derive a new expression for the q-analogue of the Young symmetrizer which generate irreducible representations of the Hecke algebra. We obtain this new expression using Cherednik's fusion procedure. However, instead of splitting Young diagrams into their rows or columns, we consider their principal hooks. This minimises the number of auxiliary parameters needed in the fusion procedure.
We present a bijective proof of the hook-length formula for shifted standard tableaux of a fixed shape based on a modified jeu de taquin and the ideas of the bijective proof of the hook-length formula for ordinary standard tableaux by Novelli, Pak and Stoyanovskii. In their proof Novelli, Pak and Stoyanovskii define a bijection between arbitrary fillings of the Ferrers diagram with the integers $1,2,...,n$ and pairs of standard tableaux and hook tabloids. In our shifted version of their algorithm the map from the set of arbitrary fillings of the shifted Ferrers diagram onto the set of shifted …
A {\em pointed partition} of $n$ is a pair $(λ, v)$ where $λ\vdash n$ and $v$ is a cell in its Ferrers diagram. We construct an involution on pointed partitions of $n$ exchanging "hook length" and "part length". This gives a bijective proof of a recent result of Bessenrodt and Han.
The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of $t$-hooks. It is well-known that the hook lengths of multiples of $t$ can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes …
We give new bounds and asymptotic estimates on the number of standard Young tableaux of skew shape in a variety of special cases. Our approach is based on Naruse's hook-length formula. We also compare our bounds with the existing bounds on the numbers of linear extensions of the corresponding posets.
We define a fat staircase to be a Ferrers diagram corresponding to a partition of the form $(n^{α_n}, {n-1}^{α_{n-1}},..., 1^{α_1})$, where $α= (α_1,...,α_n)$ is a composition, or the $180^\circ$ rotation of such a diagram. We look at collections of skew diagrams consisting of a fixed fat staircase augmented with all hooks of a given size. Among these diagrams we determine precisely which pairs give a Schur-positive difference. We extend this classification to collections of fat staircases augmented with hook-complements.
Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P \setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides…
We show that the distribution of the major index over the set of involutions in S_n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside an n/2 by n/2 box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.
Representation theory of the symmetric group $\mathfrak{S}_n$ has a very distinctive combinatorial flavor. The conjugacy classes as well as the irreducible characters are indexed by integer partitions $λ\vdash n$. We introduce class functions on $\mathfrak{S}_n$ that count the number of certain tilings of Young diagrams. The counting interpretation gives a uniform expression of these class functions in the ring of character polynomials, as defined by \cite{murnaghanfirst}. A modern treatment of character polynomials is given in \cite{orellana-zabrocki}. We prove a relation between these combin…
The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. …
We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.
Using a 0/1 encoding of Young diagrams and its consequences for rim hook tableaux, we prove a reduction formula of Littlewood for arbitrary characters of the symmetric group, evaluated at elements with all cycle lengths divisible by a given integer. As an application, we find explicitly the coefficients in a formula of Kostant for certain powers of the Dedekind $η$-function, avoiding most of the original machinery.
We study the Pieri type formulas for the Schur multiple zeta functions along with those for the Schur polynomials. To formulate these formulas, we introduce a new insertion rule for adding boxes in the Young tableaux and obtain the results for the hook type Schur multiple zeta functions. For the proof, we show {certain} extended Jacobi-Trudi formulas for the Schur multiple zeta functions.
In this paper we construct for a given arbitrary skew diagram A all partitions nu with maximal principal hook lengths among all partitions with the character [nu] appearing in the skew character [A]. Furthermore we show that these are also partitions with minimal Durfee size. This we use to give the maximal Durfee size for [nu] appearing in [A] for the cases when A decays into two partitions and for some special cases of A. Also this gives conditions for two skew diagrams to represent the same skew character.
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.
The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman--Grassl correspondence, respectively. The main new results are two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting rever…
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…
By considering the specialisation $s_λ(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $λ$ in terms of two properties of the boxes in the diagram for $λ$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $λ$-tableaux.
We study the class of functions on the set of (generalized) Young diagrams arising as the number of embeddings of bipartite graphs. We give a criterion for checking when such a function is a polynomial function on Young diagrams (in the sense of Kerov and Olshanski) in terms of combinatorial properties of the corresponding bipartite graphs. Our method involves development of a differential calculus of functions on the set of generalized Young diagrams.
We extend the polynomial approach to hook length formula proposed in a recent joint paper with Károlyi, Nagy and Volkov to several other problems of the same type, including number of paths formula in the Young graph of strict partitions.
We give a combinatorial formula for LLT cumulants of unicellular unilevelled shapes in terms of parking functions and Cayley trees. Our formula implies previously conjectured Schur positivity.
Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the $2k$-th power sum of hook lengths of partitions with size $n$ is always a polynomial of $n$ for any $k\in \mathbb{N}$. This conjecture was generalized and proved by Stanley (Ramanujan J., 23(1--3): 91--105, 2010). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators $D$ and $D^-$ defined on functions of partitions. Even though the calculations for higher orders…
The pattern-avoiding fillings of Young diagrams we study arose from Postnikov's work on positive Grassman cells. They are called Le-diagrams, and are in bijection with decorated permutations. Other closely-related diagrams are interpreted as acyclic orientations of some bipartite graphs. The definition of the diagrams is the same but the avoided patterns are different. We give here bijections proving that the number of pattern-avoiding filling of a Young diagram is the same, for these two different sets of patterns. The result was obtained by Postnikov via a reccurence relation. This relation …
Using a modified version of jeu de taquin, Novelli, Pak and Stoyanovskii gave a bijective proof of the hook-length formula for counting standard Young tableaux of fixed shape. In this paper we consider a natural extension of jeu de taquin to arbitrary posets. Given a poset P, jeu de taquin defines a map from the set of bijective labelings of the poset elements with $\{1,2,...,|P|\}$ to the set of linear extensions of the poset. One question of particular interest is for which posets this map yields each linear extension equally often. We analyze the double-tailed diamond poset $D_{m,n}$ and sh…
We determine the most general form of a smooth function on Young diagrams, that is, a polynomial in the interlacing or multirectangular coordinates whose value depends only on the shape of the diagram. We prove that the algebra of such functions is isomorphic to quasi-symmetric functions, and give a noncommutative analog of this result.
In this paper, we present the results of a computer investigation of asymptotics for maximum dimensions of linear and projective representations of the symmetric group. This problem reduces to the investigation of standard and strict Young diagrams of maximum dimensions. We constructed some sequences for both standard and strict Young diagrams with extremely large dimensions. The conjecture that the limit of normalized dimensions exists was proposed 30 years ago [A.~M.~Vershik and S.~V.~Kerov, 1985] and has not been proved yet. We studied the growth and oscillations of the normalized dimension…
We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even pol…
There exists a well-known hook-length formula for calculating the dimensions of 2D Young diagrams. Unfortunately, the analogous formula for 3D case is unknown. We introduce an approach for calculating the estimations of dimensions of three-dimensional Young diagrams also known as plane partitions. The most difficult part of this task is the calculation of co-transition probabilities for a central Markov process. We propose an algorithm for approximate calculation of these probabilities. It generates numerous random paths to a given diagram. In case the generated paths are uniformly distributed…
We study the Young graph with edge multiplicities arising in a Pieri-type formula for Jack symmetric polynomials $P_μ(x;a)$ with a parameter $a$. Starting with the empty diagram, we define recurrently the `dimensions' $\dim_a$ in the same way as for the Young lattice or Pascal triangle. New proofs are given for two known results. The first is the $a$-hook formula for $\dim_a$, first found by R.Stanley. Secondly, we prove (for all complex $u$ and $v$) a generalization of the identity $\sumν(c(b)+u)(c(b)+v)\dimν/\dimμ=(n+1)(n+uv)$, where $ν$ runs over immediate successors of a Young diagram $μ$ …
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…
