|
In combinatorial mathematics, an alternating permutation of the set is an arrangement of those numbers into an order ''c''1, ..., ''c''''n'' such that no element ''c''''i'' is between ''c''''i'' − 1 and ''c''''i'' + 1 for any value of ''i'' and ''c''1< ''c''2. In other words, ''c''''i'' < ''c''''i''+ 1 if ''i'' is odd and ''c''''i'' > ''c''''i''+ 1 if ''i'' is even. For example, the five alternating permutations of are: * 1, 3, 2, 4 because 1 < 3 > 2 < 4 * 1, 4, 2, 3 because 1 < 4 > 2 < 3 * 2, 3, 1, 4 because 2 < 3 > 1 < 4 * 2, 4, 1, 3 because 2 < 4 > 1 < 3 * 3, 4, 1, 2 because 3 < 4 > 1 < 2 This type of permutation was first studied by Désiré André in the 19th century.〔Jessica Millar, N. J. A. Sloane, Neal E. Young, ("A New Operation on Sequences: the Boustrouphedon Transform" ) J. Combinatorial Theory, Series A 76(1):44–54 (1996)〕 If the condition ''c''1 < ''c''2 is dropped, so we only require that no element ''c''''i'' is between ''c''''i'' − 1 and ''c''''i'' + 1, then the permutation is called a zigzag permutation. By exchanging 1 with ''n'', 2 with ''n'' − 1, etc., each zigzag permutation with ''c''1> ''c''2 can be paired uniquely with an alternating permutation. ==Related integer sequences== The determination of the number, ''A''''n'', of alternating permutations of the set is called André's problem. If ''Z''''n'' denotes the number of zigzag permutations of then it is clear from the pairing given above that ''Z''''n'' = 2''A''''n'' for ''n'' ≥ 2. The numbers ''A''''n'' are known as Euler zigzag numbers or up/down numbers. The first few values of ''A''''n'' are 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, ... . The first few values of ''Z''''n'' are 1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042, ... . The Euler zigzag numbers are related to Entringer numbers, from which the zigzag numbers may be computed. The ''n''th zigzag number is equal to the Entringer number ''E''(''n'', ''n''), and these Entringer numbers can be defined recursively as follows:〔Weisstein, Eric W. "Entringer Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EntringerNumber.html〕 : : : . The numbers ''A''2''n'' with even indices are called secant numbers or zig numbers. The first few values are 1, 1, 5, 61, 1385, 50521, ... . They appear as numerators in the Maclaurin series of sec ''x''. Specifically, : Secant numbers are related to Euler numbers by the formula ''E''2''n'' = (−1)''n''''A''2''n''. (''E''''n'' = 0 when ''n'' is odd.) Correspondingly, the numbers ''A''2''n''+1 with odd indices are called tangent numbers or zag numbers. The first few values are 1, 2, 16, 272, 7936, ... . They appear as numerators in the Maclaurin series of tan ''x''. Specifically, : Tangent numbers are related to Bernoulli numbers by the formula : for ''n'' > 0. Adding these series together gives the exponential generating function of the sequence ''A''''n'': : : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「alternating permutation」の詳細全文を読む スポンサード リンク
|