<abstract xmlns="http://www.w3.org/1999/xhtml"><p> 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. </p></abstract>

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.

Some identities for derangement and Ward number sequences and related bijections

Department of Statistics University of Wisconsin Madison, WI 53706, USA

Department of Mathematics University of Haifa 3498838 Haifa, Israel

Department of Mathematics University of Tennessee Knoxville, TN 37996 USA

Pure Mathematics and Applications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Some identities for derangement and Ward number sequences and related bijections"}

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:...

Some identities for derangement and Ward number sequences and related bijections

Published Online: Jun 24, 2016

Page range: 132 - 143

Received: Jan 07, 2016

DOI: https://doi.org/10.1515/puma-2015-0013

Keywords
derangement, Ward numbers, Stirling numbers, combinatorial proof

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Some identities for derangement and Ward number sequences and related bijections

Published Online: Jun 24, 2016

Page range: 132 - 143

Received: Jan 07, 2016

DOI: https://doi.org/10.1515/puma-2015-0013

Keywordsderangement, Ward numbers, Stirling numbers, combinatorial proof

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
derangement, Ward numbers, Stirling numbers, combinatorial proof