Reciprocity for multirestricted Stirling numbers

Authors:
Ji Young Choi;Ling Long;Siu-Hung Ng;Jonathan Smith
Affiliations:
Department of Mathematics, Shippensburg University, Shippensburg, PA;Department of Mathematics, Iowa State University, Ames, IA;Department of Mathematics, Iowa State University, Ames, IA;Department of Mathematics, Iowa State University, Ames, IA
Venue:
Journal of Combinatorial Theory Series A
Year:
2006

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Abstract

Multirestricted Stirling numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most a fixed number of elements. Multirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of corresponding multirestricted numbers of the second kind. The anomalous sign behavior of these latter numbers makes them impervious to combinatorial analysis. In answer to a conjecture that has remained open for several years, we derive a reciprocity law for multirestricted Stirling numbers using algebraic techniques based on polynomial recursions. As corollaries, we obtain new recurrence relations for multirestricted numbers, and a new algebraic derivation of the reciprocity law for Stirling numbers.