Riv. Mat. Univ. Parma, Vol. 11, No. 2, 2020

Margherita Maria Ferrari [a], Emanuele Munarini [b] and Norma Zagaglia Salvi [b]

Some combinatorial properties of the generalized derangement numbers

Pages: 217-249
Received: 9 April 2019
Accepted in revised form: 27 August 2020
Mathematics Subject Classification (2010): Primary 05A19, 05A15; Secondary 11B73, 11B83.
Keywords: species, permutation, derangement, arrangement, enriched partition, enriched partition with no singleton block, rencontres polynomial, Stirling number, Bell number.
Authors address:
[a]: Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Avenue, Tampa, FL 33620, USA
[b]: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

This research was partially supported by the NSF CCF-1526485 and DMS-1800443, the NIH R01GM109459-01, the Southeast Center for Mathematics and Biology (an NSF-Simons Research Center for Mathematics of Complex Biological Systems) under NSF DMS-1764406 and Simons Foundation 594594 (first author).

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Abstract: In this paper, we give a simple description of the \(m\)-widened permutations (generalized \(m\)-permutations) and the \(m\)-widened derangements (generalized \(m\)-derangements) in terms of ordinary permutations and derangements with a suitable constraint. This approach allows us to give a natural combinatorial interpretation of the generalized derangement numbers and the generalized rencontres polynomials in terms of species of structures. Finally, we obtain some formulas relating the generalized derangement numbers with the \(r\)-Bell numbers. In particular, we give an extension of the Clarke-Sved identity.

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