Rook numbers and the normal ordering problem

2004/02/23 by Anna Varvak, Varvak, Anna · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #math.CO #msc:05A10

paper · pdf · doi:10.48550/arxiv.math/0402376

14 pages, presented as poster in FPSAC'04

arxiv created 2004/07/15 · arxiv updated 2009/12/01

Abstract

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑ c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend all these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.

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