A Randomq, t-Hook Walk and a Sum of Pieri Coefficients

Abstract

This work deals with the identityBμ(q, t)=∑ν→μcμν(q, t), whereBμ(q, t) denotes the biexponent generator of a partitionμ. That is,Bμ(q, t)=∑s∈μqa′(s)tl′(s), witha′(s) andl′(s) the co-arm and co-leg of the lattice squaresinμ. The coefficientscμν(q, t) are closely related to certain rational functions occuring in one of the Pieri rules for the Macdonald polynomials and the symbolν→μis used to indicate that the sum is over partitionsνwhich immediately precedeμin the Young lattice. This identity has an indirect manipulatorial proof involving a number of deep identities established by Macdonald. We show here that it may be given an elementary probabilistic proof by a mechanism which emulates the Greene–Nijehuis–Wilf proof of the hook formula.