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Turkish Journal of Mathematics

Authors

WESAM TALAB

Abstract

In this article, we study word standardization in comparison to Young tableau standardization. We count the number of words (respectively Young tableau) standardized to a given permutation (respectively to a given standard Young tableau). We prove that both rectification and standardization applications commute and show that the standardization commutes with the insertion of Robinson--Schensted. We show that the standardizations of Knuth-equivalent two words are also Knuth equivalent. Finally, using word standardization we establish a proof for the following well-known equality: $$ \forall l \in \left\lbrace 0,1,\ldots,n-1\right\rbrace ,~~\left \langle {n\atop l} \right \rangle=d_{n,l}=a_{n,l}= \sum_{0\leq k \leq l}(-1)^k { n+1 \choose k } (l+1-k)^n. $$

DOI

10.55730/1300-0098.3248

Keywords

RSK the correspondence of Robinson-Schensted-Knuth, Young tableaux, word standardization, Knuth equivalent of words, Eulerian number

First Page

2003

Last Page

2014

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Mathematics Commons

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