scholarly journals Hook lengths in a skew Young diagram

10.37236/1309 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Young Diagram ◽  
Character Degrees ◽  
Young Diagrams ◽  
Inductive Proof

Regev and Vershik (Electronic J. Combinatorics 4 (1997), #R22) have obtained some properties of the set of hook lengths for certain skew Young diagrams, using asymptotic calculations of character degrees. They also conjectured a stronger form of one of their results. We give a simple inductive proof of this conjecture. Very recently, Regev and Zeilberger (Annals of Combinatorics, to appear) have independently proved this conjecture.

scholarly journals Search for Young diagrams with large dimensions

2019 ◽  
pp. 33-43
Author(s):  
Vasilii S. Duzhin ◽  
◽  
Anastasia A. Chudnovskaya ◽  
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Numerical Experiments ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Maximum Dimension ◽  
Asymptotic Combinatorics

Search for Young diagrams with maximum dimensions or, equivalently, search for irreducible representations of the symmetric group $S(n)$ with maximum dimensions is an important problem of asymptotic combinatorics. In this paper, we propose algorithms that transform a Young diagram into another one of the same size but with a larger dimension. As a result of massive numerical experiments, the sequence of $10^6$ Young diagrams with large dimensions was constructed. Furthermore, the proposed algorithms do not change the first 1000 elements of this sequence. This may indicate that most of them have the maximum dimension. It has been found that the dimensions of all Young diagrams of the resulting sequence starting from the 75778th exceed the dimensions of corresponding diagrams of the greedy Plancherel sequence.

scholarly journals Permutations Without Long Decreasing Subsequences and Random Matrices

10.37236/929 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Piotr Šniady
Keyword(s):  
Random Matrices ◽  
Young Diagram ◽  
Random Matrix ◽  
Random Permutation ◽  
Fixed Number ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Gaussian Unitary Ensemble ◽  
Longest Increasing Subsequence ◽  
Unitary Ensemble

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

scholarly journals Osculating Paths and Oscillating Tableaux

10.37236/731 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Roger E. Behrend
Keyword(s):  
Young Diagram ◽  
Fixed Number ◽  
Lattice Points ◽  
Lattice Paths ◽  
Square Lattice ◽  
Vertex Model ◽  
Young Diagrams ◽  
Primary Interest ◽  
Alternating Sign Matrices ◽  
Positive Integers

The combinatorics of certain tuples of osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. The paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths within a tuple are permitted to share lattice points, but not to cross or share lattice edges. Such path tuples correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices. Of primary interest here are path tuples with a fixed number $l$ of vacancies and osculations, where vacancies or osculations are points of the rectangle through which respectively no or two paths pass. It is shown that there exist natural bijections which map each such path tuple $P$ to a pair $(t,\eta)$, where $\eta$ is an oscillating tableau of length $l$ (i.e., a sequence of $l+1$ partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and $t$ is a certain, compatible sequence of $l$ weakly increasing positive integers. Furthermore, each vacancy or osculation of $P$ corresponds to a partition in $\eta$ whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for tuples of osculating paths involving sums over oscillating tableaux.

scholarly journals Characters of symmetric groups in terms of free cumulants and Frobenius coordinates

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Maciej Dolega ◽  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Young Diagram ◽  
Symmetric Groups ◽  
Main Tool ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Free Cumulants ◽  
International Audience

International audience Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups $\mathfrak{S}(n)$ in the limit $n \to \infty$. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams. Les cumulants libres sont des fonctions agréables et utiles sur l'ensemble des diagrammes de Young, en particulier, ils donnent le comportement asymptotiques des caractères normalisés du groupe symétrique $\mathfrak{S}(n)$ dans la limite $n \to \infty$. Nous donnons une formule combinatoire explicite pour les caractères normalisés du groupe symétrique en fonction des cumulants libres. Nous exprimons également les caractères en fonction des coordonnées de Frobenius. Nos formules font intervenir le nombre de certaines factorisations d'une permutation donnée. L'outil principal est la famille de polynômes de Stanley donnant les valeurs des caractères sur les diagrammes de Young multirectangulaires.

Star Diagrams and the Symmetric Group

1950 ◽  
Vol 2 ◽  
pp. 79-92 ◽  
Author(s):  
R. A. Staal
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
One To One

Introduction. The irreducible representations of the symmetric group Sn, were shown by A. Young to be in one-to-one correspondence with certain arrays of n nodes. E.g. for n = 12 and the partition λ = [4, 4, 3, 1] we have the array which we call a “Young diagram.” The question arises as to the manner in which various properties of the representations are reflected in their corresponding Young diagrams.

scholarly journals Shape-Wilf-Ordering on Permutations of Length 3

10.37236/974 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Zvezdelina Stankova
Keyword(s):  
Young Diagram ◽  
Pattern Avoidance ◽  
Current Paper ◽  
Young Diagrams ◽  
Arbitrary Length ◽  
Limited Knowledge

The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits $\lim_{n\rightarrow \infty} \sqrt[n]{|S_n(\tau)|}$ and further works suggest asymptotic ordering of layered versus monotone patterns. Yet, Bóna has provided essentially the only known up to now result of its type on complete ordering of $S_k$ for $k=4$: $|S_n(1342)| < |S_n(1234)| < |S_n(1324)|$ for $n\geq 7$, along with some other sporadic examples in Wilf-ordering. We give a different proof of this result by ordering $S_3$ up to the stronger shape-Wilf-order: $|S_Y(213)|\leq |S_Y(123)|\leq |S_Y(312)|$ for any Young diagram $Y$, derive as a consequence that $|S_Y(k+2,k+1,k+3,\tau)|\leq |S_Y(k+1,k+2,k+3,\tau)|\leq |S_Y(k+3,k+1,k+2,\tau)|$ for any $\tau\in S_k$, and find out when equalities are obtained. (In particular, for specific $Y$'s we find out that $|S_Y(123)|=|S_Y(312)|$ coincide with every other Fibonacci term.) This strengthens and generalizes Bóna's result to arbitrary length permutations. While all length-3 permutations have been shown in numerous ways to be Wilf-equivalent, the current paper distinguishes between and orders these permutations by employing all Young diagrams. This opens up the question of whether shape-Wilf-ordering of permutations, or some generalization of it, is not the "true" way of approaching pattern-avoidance ordering.

scholarly journals Poset Pinball, Highest Forms, and $(n-2,2)$ Springer Varieties

10.37236/2126 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Barry Dewitt ◽  
Megumi Harada
Keyword(s):  
Young Diagram ◽  
Equivariant Cohomology ◽  
Type A ◽  
Linear Operators ◽  
Young Diagrams ◽  
Adjacent Pair ◽  
Open Questions ◽  
Hessenberg Varieties ◽  
Study Type ◽  
Future Work

In this manuscript we study type $A$ nilpotent Hessenberg varieties equipped with a natural $S^1$-action using techniques introduced by Tymoczko, Harada-Tymoczko, and Bayegan-Harada, with a particular emphasis on a special class of nilpotent Springer varieties corresponding to the partition $\lambda= (n-2,2)$ for $n \geq 4$. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with $n$ boxes with the alphabet $\{1,2,\ldots,n\}$. Using the adjacent-pair matrix we make more explicit and also extend some statements concerning highest forms of linear operators in previous work of Tymoczko. Second, for a nilpotent operator $N$ and Hessenberg function $h$, we construct an explicit bijection between the $S^1$-fixed points of the nilpotent Hessenberg variety Hess$(N,h)$ and the set of $(h,\lambda_N)$-permissible fillings of the Young diagram $\lambda_N$. Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the $S^1$-equivariant cohomology of type $A$ Springer varieties $\mathcal{S}_{(n-2,2)}$ associated to Young diagrams of shape $(n-2,2)$ for $n\geq 4$. Specifically, we use the dimension pair algorithm for Betti-acceptable pinball described by Bayegan and Harada to specify a subset of the equivariant Schubert classes in the $\mathbb{T}$-equivariant cohomology of the flag variety $\mathcal{F}\ell ags(\mathbb{C}^n) \cong GL(n,\mathbb{C})/B$ which maps to a module basis of $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$ under the projection map $H^*_\mathbb{T}(\mathcal{F}\ell ags(\mathbb{C}^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. Our poset pinball module basis is not poset-upper-triangular; this is the first concrete such example in the literature. A straightforward consequence of our proof is that there exists a simple and explicit change of basis which transforms our poset pinball basis to a poset-upper-triangular module basis for $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. We close with open questions for future work.

scholarly journals Neighborhood Growth Dynamics on the Hamming Plane

10.37236/6400 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Janko Gravner ◽  
David Sivakoff ◽  
Erik Slivken
Keyword(s):  
Young Diagram ◽  
Growth Dynamics ◽  
Vertical Line ◽  
Product Measure ◽  
Two Dimensional ◽  
Young Diagrams ◽  
Entire Plane ◽  
Initial Product ◽  
Entropy Functional ◽  
Hamming Graphs

We initiate the study of general neighborhood growth dynamics on two-dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram.  We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams.

Some combinatorial results involving Young diagrams

1978 ◽  
Vol 83 (1) ◽  
pp. 1-10 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
New Method ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Sufficient Criterion ◽  
Power Diagram ◽  
Necessary And Sufficient

In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

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