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Alternating subsets and permutations. (English) Zbl 1206.05006

Summary: We give new proofs of theorems on alternating subsets of integers by means of bijective transformations. It is shown that all known results are consequences of a simple result on the residue class of an integer. The notion of alternating subset is extended to permutations of \(\{1,2,\dots, n\}\). In particular, we obtain solutions to the problems of Terquem and Skolem’s generalization for permutations.

MSC:

05A05 Permutations, words, matrices
11B50 Sequences (mod \(m\))
05A15 Exact enumeration problems, generating functions

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Online Encyclopedia of Integer Sequences:

a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1.

References:

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