A bijective proof of Loehr-Warrington's formulas for the statistics $\mbox{ctot}_{\frac{q}{p}}$ and $\mbox{midd}_{\frac{q}{p}}$

Abstract

Loehr and Warrington introduced partitional statistics $\mbox{ctot}_{\frac{q}{p}}(D)$ and $\mbox{midd}_{\frac{q}{p}}(D)$ and provided formulas for these statistics in terms of the boundary graph of the Young diagram $D$. In this paper we give a bijective proof of Loehr-Warrington's formulas using the following simple combinatorial observation: given a Young diagram $D$ and two numbers $a$ and $l,$ the number of boxes in $D$ with the arm length $a$ and the leg length $l$ is one less than the number of boxes with the same properties in the complement to $D.$ Here the complement is taken inside the positive quadrant or, equivalently, a very large rectangle.