Variations of the numbers of permutations and tableaux

Abstract

Let $f^{\lambda}$ be the number of standard Young tableaux of shape $\lambda$. By Robinson-Schensted correspondence we have $\sum_{\lambda \vdash n} (f^{\lambda})^2 = n!,$ (1) $\sum_{\lambda \vdash n} f^{\lambda} = t_n,$ (2) where $t_n$ denotes the number of involutions of length $\textit{n}$. For a SYT $\textsl{T}$, the sign of $\textsl{T}$ is defined by sign$(\pi)$, where $\pi$ is the permutation obtained by reading $\textsl{T}$ like a book. For example, if $\textsl{T}$ = $\psraise (2,1){\pspicture (0,-2) (3,0) \cell(1,1)[1] \cell(1,2)[2] \cell(1,3)[4] \cell(2,1)[3] \cell(2,2)[5] \endpspicture}$ then sign ($\textsl{T}$) = sign(12435) = -1. The sign-imbalance $I_{\lambda}$ of a partition $\lambda$ is the sum of $\textsl(T)$ for all SYTs $\textsl{T}$ of shape $\lambda$. Stanley suggested interesting sign-imbalance formulas which are sign variations of (1) and (2). The simplest forms are the following: $\sum_{\lambda \vdash n} (-1)^{v(\lambda)}I_{\lambda}^2 = 0$, (3) $\sum_{\lambda \vdash n} I_{\lambda} = 2^{{\left \lfloor \frac{n}{2} \right \rfloor}}$, (4) where $v(\lambda)$ denotes the sum of even parts of $\lambda$. The aim of this thesis is to study variations of (1), (2), (3) and (4)