Robinson-Schensted Algorithms Obtained from Tableau Recursions

Abstract

The numbers $f_\lambda$ of standard tableaux of shape $\lambda\vdash n$ satisfy 2 fundamental recursions: $f_\lambda = \sum f_{\lambda^-}$ and $(n + 1)f_\lambda=\sum f_{\lambda^+}$, where $\lambda^-$ and $\lambda^+$ run over all shapes obtained from $\lambda$ by adding or removing a square respectively. The first of these recursions is trivial; the second can be proven algebraically from the first. These recursions together imply algebraically the dimension formula $n! =\sum f_\lambda^2$ for the irreducible representations of $S_n$. We show that a combinatorial analysis of this classical algebraic argument produces an infinite family of algorithms, among which are the classical Robinson-Schensted row and column insertion algorithms. Each of our algorithms yields a bijective proof of the dimension formula.