Chemins sous-diagonaux et tableaux de Young

Abstract

We consider path in the lattice of positive integer coordinate where the possible "moves" are of four kinds : (1) increasing the x coordinate by 1, (2) decreasing the x coordinate by 1, (3) increasing the y coordinate by 1, (4) decreasing the y coordinate by 1. The number of such paths of length ℓ, from (0,0) to any point whose y-coordinate is 0, lying below or touching the main diagonal, is CnCn+1 for ℓ=2n and Cn+1Cn+1 for ℓ=2n+1 where Cn is the Catalan number. We give a bijective proof of this result. As corollary we give exact formulas for the number of standard Young tableaux having n cells and a most k rows in the cases k=4 and k=5.KeywordsCatalan NumberStandard Young TableauBijective ProofColumn Strict TableauBinomial DeterminantThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.