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

✍ Scribed by A.M Garsia; M Haiman

• Book ID: 102583153
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 488 KB
• Volume: 82
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2842

✦ Synopsis

This work deals with the identity B + (q, t)= & Ä + c +& (q, t), where B + (q, t) denotes the biexponent generator of a partition +. That is, B + (q, t)= s # + q a$(s) t l $(s) , with a$(s) and l $(s) the co-arm and co-leg of the lattice square s in +. The coefficients c +& (q, t) are closely related to certain rational functions occuring in one of the Pieri rules for the Macdonald polynomials and the symbol & Ä + is used to indicate that the sum is over partitions & which immediately precede + in 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.