Munagi, Augustine O. Alternating subsets and permutations. (English) Zbl 1206.05006 Rocky Mt. J. Math. 40, No. 6, 1965-1977 (2010). 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. Cited in 4 Documents MSC: 05A05 Permutations, words, matrices 11B50 Sequences (mod \(m\)) 05A15 Exact enumeration problems, generating functions Keywords:alternating subset; terquem problem; permutation Software:OEIS PDFBibTeX XMLCite \textit{A. O. Munagi}, Rocky Mt. J. Math. 40, No. 6, 1965--1977 (2010; Zbl 1206.05006) Full Text: DOI Online Encyclopedia of Integer Sequences: a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1. References: [1] M. Abramson and W. Moser, Generalizations of Terquem’s problem , J. Combin. Theory 7 (1969), 171-180. · Zbl 0181.02103 [2] L. Carlitz, Alternating sequences , Discrete Math. 17 (1977), 133-138. · Zbl 0361.05003 [3] C.A. Church and H.W. Gould, Lattice point solution of the generalized problem of Terquem and an extension of Fibonacci numbers , Fibonacci Quart. 5 (1967), 59-68. · Zbl 0161.01202 [4] I.P. Goulden and D.M. Jackson, The enumeration of generalized alternating subsets with congruences , Discrete Math. 22 (1978), 99-104. · Zbl 0392.05006 [5] F. Hering, A problem of inequalities , Amer. Math. Monthly 78 (1971), 275-276. JSTOR: [6] E. Netto, Lehrbuch der Combinatorik , 2nd ed., Leipzig, 1927, reprinted by Chelsea, New York, 1958. · JFM 53.0073.09 [7] J. Riordan, An introduction to combinatorial analysis , Wiley, New York, 1958. · Zbl 0078.00805 [8] D. Singmaster, Problem 1654, Math. Magazine 75 (October 2002), Solution in Math. Magazine 76 (October 2003). [9] N.J.A. Sloane, The on-line encyclopedia of integer sequences , published electronically at www.research.att.com/ njas/sequences/, 2007. [10] S.M. Tanny, On alternating subsets of integers , Fibonacci Quart. 13 (1975), 325-328. · Zbl 0327.05011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.