7 research outputs found
Some identities on derangement and degenerate derangement polynomials
In combinatorics, a derangement is a permutation that has no fixed points.
The number of derangements of an n-element set is called the n-th derangement
number. In this paper, as natural companions to derangement numbers and
degenerate versions of the companions we introduce derangement polynomials and
degenerate derangement polynomials. We give some of their properties,
recurrence relations and identities for those polynomials which are related to
some special numbers and polynomials.Comment: 12 page
Cyclic derangements
A classic problem in enumerative combinatorics is to count the number of
derangements, that is, permutations with no fixed point. Inspired by a recent
generalization to facet derangements of the hypercube by Gordon and McMahon, we
generalize this problem to enumerating derangements in the wreath product of
any finite cyclic group with the symmetric group. We also give q- and (q,
t)-analogs for cyclic derangements, generalizing results of Brenti and Gessel.Comment: 14 page
Naturally emerging maps for derangements and nonderangements
A derangement is a permutation with no fixed point, and a nonderangement is a
permutation with at least one fixed point. There is a one-term recurrence for
the number of derangements of elements, and we describe a bijective proof
of this recurrence which can be found using a recursive map. We then show the
combinatorial interpretation of this bijection and how it compares with other
known bijections, and show how this gives an involution on .
Nonderangements satisfy a similar recurrence. We convert the bijective proof of
the one-term identity for derangements into a bijective proof of the one-term
identity for nonderangements.Comment: 17 page