scholarly journals A Combinatorial Formula for the Hilbert Series of Bigraded $S_n$-Modules

10.37236/365 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Meesue Yoo
Keyword(s):  
Hilbert Series ◽  
Macdonald Polynomials ◽  
Weighted Sums ◽  
Young Tableaux ◽  
Combinatorial Formula ◽  
Standard Young Tableaux

We prove a combinatorial formula for the Hilbert series of the Garsia-Haiman bigraded $S_n$-modules as weighted sums over standard Young tableaux in the hook shape case. This method is based on the combinatorial formula of Haglund, Haiman and Loehr for the Macdonald polynomials and extends the result of A. Garsia and C. Procesi for the Hilbert series when $q=0$. Moreover, we construct an association of the fillings giving the monomial terms of Macdonald polynomials with the standard Young tableaux.

scholarly journals Combinatorial Formula for the Hilbert Series of bigraded $S_n$-modules

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Meesue Yoo
Keyword(s):  
Hilbert Series ◽  
Macdonald Polynomials ◽  
Weighted Sum ◽  
Young Tableaux ◽  
Combinatorial Formula ◽  
Nous Proposons ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Littlewood Polynomial ◽  
The Way

International audience We introduce a combinatorial way of calculating the Hilbert series of bigraded $S_n$-modules as a weighted sum over standard Young tableaux in the hook shape case. This method is based on Macdonald formula for Hall-Littlewood polynomial and extends the result of $A$. Garsia and $C$. Procesi for the Hilbert series when $q=0$. Moreover, we give the way of associating the fillings giving the monomial terms of Macdonald polynomials to the standard Young tableaux. Nous introduisons une méthode combinatoire pour calculer la série de Hilbert de modules bigradués de $S_n$ comme une somme pondérée sur les tableaux de Young standards à la forme crochet. Cette méthode se fonde sur la formule Macdonald pour les polynômes Hall-Littlewood et généralise un résultat de $A$. Garsia et $C$. Procesi pour la série de Hilbert dans le cas $q=0$. De plus, nous proposons une méthode pour associer aux tableaux de Young standards les remplissages des monômes des polynômes de Macdonald.

scholarly journals Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices

10.37236/6888 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Martha Yip
Keyword(s):  
Finite Field ◽  
Weighted Sum ◽  
Young Tableaux ◽  
Canonical Forms ◽  
Combinatorial Formula ◽  
Rook Placements ◽  
Nilpotent Matrices ◽  
Jordan Forms ◽  
Standard Young Tableaux ◽  
Young’S Lattice

The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $\mathbb{F}_q$ has Jordan canonical forms indexed by partitions $\lambda \vdash n$. We present a combinatorial formula for computing the number $F_\lambda(q)$ of matrices of Jordan type $\lambda$ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for $F_\lambda(q)$.

Standard Young tableaux in a (2,1)-hook and Motzkin paths

2021 ◽  
Vol 344 (7) ◽  
pp. 112395
Author(s):  
Rosena R.X. Du ◽  
Jingni Yu
Keyword(s):  
Young Tableaux ◽  
Standard Young Tableaux ◽  
Motzkin Paths

scholarly journals A combinatorial formula for nonsymmetric Macdonald polynomials

2008 ◽  
Vol 130 (2) ◽  
pp. 359-383 ◽  
Author(s):  
James. Haglund ◽  
Mark D. Haiman ◽  
N. Loehr
Keyword(s):  
Macdonald Polynomials ◽  
Combinatorial Formula

scholarly journals A direct bijective proof of the hook-length formula

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Igor Pak ◽  
Alexander V. Stoyanovskii
Keyword(s):  
Young Tableaux ◽  
Hook Length ◽  
Bijective Proof ◽  
International Audience ◽  
Standard Young Tableaux

International audience This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

scholarly journals A Combinatorial Approach to the $q,t$-Symmetry Relation in Macdonald Polynomials

10.37236/5350 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Maria Monks Gillespie
Keyword(s):  
Macdonald Polynomials ◽  
Combinatorial Proof ◽  
Symmetry Relation ◽  
Combinatorial Approach ◽  
Combinatorial Formula ◽  
Recursive Structure ◽  
Littlewood Polynomials

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t)=\widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $\mu$. We also provide a proof for the full relation in the case when $\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

scholarly journals Enumeration of Standard Young Tableaux of Shifted Strips with Constant Width

10.37236/6466 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Integral Method ◽  
Recurrence Relations ◽  
Young Tableau ◽  
Constant Width ◽  
Young Tableaux ◽  
Pell Number ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

Let $g_{n_1,n_2}$ be the number of standard Young tableau of truncated shifted shape with $n_1$ rows and $n_2$ boxes in each row. By using the integral method this paper derives the recurrence relations of $g_{3,n}$, $g_{n,4}$ and $g_{n,5}$ respectively. Specifically, $g_{n,4}$ is the $(2n-1)$-st Pell number.

scholarly journals Evaluating the Numbers of some Skew Standard Young Tableaux of Truncated Shapes

10.37236/3890 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Young Tableaux ◽  
Multiple Integrals ◽  
Standard Young Tableaux ◽  
Product Formulas

In this paper the number of standard Young tableaux (SYT) is evaluated by the methods of multiple integrals and combinatorial summations. We obtain the product formulas of the numbers of skew SYT of certain truncated shapes, including the skew SYT $((n+k)^{r+1},n^{m-1}) / (n-1)^r $ truncated by a rectangle or nearly a rectangle, the skew SYT of truncated shape $((n+1)^3,n^{m-2}) / (n-2) \backslash \; (2^2)$, and the SYT of truncated shape $((n+1)^2,n^{m-2}) \backslash \; (2)$.

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