An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula

The purpose of this article is to give a bijective proof for Stanley’s hook-content formula [15, Theorem 15.3] for a certain plane partition generating function. In order to be able to state the formula we have to recall some basic notions from partition theory. A partition is a sequence = ( 1; 2; : : : ; r) with 1 2 r > 0, for some r. The Ferrers diagram of is an array of cells with r leftjustified rows and i cells in row i. Figure 1.a shows the Ferrers diagram corresponding to (4; 3; 3; 1). The conjugate of is the partition ( 01; : : : ; 0 1) where 0j is the length of the j-th column in the Ferrers diagram of . We label the cell in the i-th row and j-th column of (the Ferrers diagram of) by the pair (i; j). Also, if we write 2 we mean ‘ is a cell of ’. The hook length h of a cell = (i; j) of is ( i j) + ( 0j i) + 1, the number of cells in the hook of , which is the set of cells that are either in the same row as and to the right of , or in the same column as and below , included. The content of a cell = (i; j) of is j i. 1Supported in part by EC’s Human Capital and Mobility Program, grant CHRX-CT93-0400 and the Austrian Science Foundation FWF, grant P10191-PHY