Robinson-Schensted Algorithms Obtained from Tableau Recursions

Abstract

The numbers $f_{\lambda }$ of standard tableaux of shape $\lambda ⊢n$ 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.