Standard Young tableaux of height 4 and 5. (English) Zbl 0672.05012

The number of standard Young tableaux the shape of which has n cells or boxes and at most k rows is denoted by \(S^ k_ n\). The motivation to find expressions for \(S^ k_ n\) comes from various other areas besides combinatorics such as the theory of symmetric functions, invariant theory, algebraic geometry, the theory of polynomials identities. The Procesi-Razmyslov theory of trace identities, and the theory of algorithms.
Formulae for \(S^ 2_ n\) and \(S^ 3_ n\) are known, and the Catalan number or Motzkin number appear in these expressions. In the present paper, exact formulae for \(S^ 4_ n\) and \(S^ 5_ n\) are obtained. The proofs are combinatorial. First, a bijection between involutions and labelled Motzkin words is recalled. Then stacks of an involution are introduced, the height of which are related to the lengths of descreasing sequences in the involution. Connections between stacks and labelled Motzkin words are examined. Such considerations lead to a bijection between standard Young tableaux having at most 4 rows and pairs of non- crossing Dyck paths, thus enabling the author to obtain formulae for \(S^ 4_ n\) and \(S^ 5_ n\). Finally it is shown that the corresponding generating functions are not algebraic.
Reviewer: J.Van der Jeugt


05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions
05A15 Exact enumeration problems, generating functions
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