TY - JOUR
T1 - Some identities for derangement and Ward number sequences and related bijections
AU - Callan, David
AU - Mansour, Toufik
AU - Shattuck, Mark
PY - 2015
Y1 - 2015
N2 - We establish an alternating sum identity for three classes of singleton-free set partitions wherein the number of elements minus the number of blocks is fixed: (i) permutations, that is, partitions into cycles, (ii) unrestricted partitions, and (iii) contents-ordered partitions. Both algebraic and combinatorial proofs are given, the latter making use of a sign-changing involution in each ease. As a consequence, combinatorial proofs are found of specific cases of recent identities of Gould et al. involving both kinds of Stirling numbers.
AB - We establish an alternating sum identity for three classes of singleton-free set partitions wherein the number of elements minus the number of blocks is fixed: (i) permutations, that is, partitions into cycles, (ii) unrestricted partitions, and (iii) contents-ordered partitions. Both algebraic and combinatorial proofs are given, the latter making use of a sign-changing involution in each ease. As a consequence, combinatorial proofs are found of specific cases of recent identities of Gould et al. involving both kinds of Stirling numbers.
U2 - https://doi.org/10.1515/puma-2015-0013
DO - https://doi.org/10.1515/puma-2015-0013
M3 - مقالة
SN - 1788-800X
VL - 25
SP - 132
EP - 143
JO - Pure Mathematics and Applications
JF - Pure Mathematics and Applications
IS - 2
ER -