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In combinatorics, a derangement is a permutation that has no fixed points. The number of derangements of an n-element set is called the nth 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.

Some Identities on Derangement and Degenerate Derangement Polynomials