Rook numbers and the normal ordering problem

  • Authors:

  • Anna Varvak

  • Affiliations:

  • Soka University of America, Aliso Viejo, CA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2005

Quantified Score

Hi-index 0.00

Visualization

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 = Σci,j Ui Dj. We demonstrate that the normal order coefficients ci,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 ci,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D + U)n in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.