**Fairouz Beggas**^{[a]},
**Margherita Maria Ferrari**^{[b]} and
**Norma Zagaglia Salvi **^{[c]}

*Combinatorial Interpretations and Enumeration of Particular Bijections
*

**Pages:**

**Received:** 23 June 2015

**Accepted in revised form:** 19 July 2016

**Mathematics Subject Classification (2010):** 05A05, 05A15,
05A19.

**Keywords:** Permutation, derangement, species, linear species, permutation species, uniform species, derivative of a species, isomorphic
species.

**Author address:**

[a]: University of Lyon, LIRIS UMR5205 CNRS, Claude Bernard Lyon 1
University,
43 Bd du 11 Novembre 1918,
Villeurbanne, F-69622, France

[b], [c]: Politecnico di Milano, IDipartimento di Matematica, P.zza Leonardo da Vinci 32,
Milano, 20133, Italy

**Abstract:**
Let \(n\) be a nonnegative integer. We call *widened permutation* a bijection between two \((n+1)\)-sets
having \(n\) elements in common.
A *widened permutation* is a widened permutation without fixed points.
In this paper we determine combinatorial interpretations of these functions in
the context of the theory of species of Joyal.
In particular, we prove that the species of the widened permutations is isomorphic to
the derivative of the species of permutations. Looking at the generating series we obtain
enumerative results, which are also obtained in a direct way. Finally, we prove that the sequence
of widened derangement numbers turns out to coincide with the integer sequence A000255 of the On-Line Encyclopedia
of Integer Sequences..

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[7]
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*Enumerative combinatorics (Vol. 1)*,
Cambridge University Press, Cambridge, 1997.
MR1442260

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