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 A197365 T(n,k) gives the number of permutations of the set [n] that contain k occurrences of the subword (132). 12
 1, 1, 2, 5, 1, 16, 8, 63, 54, 3, 296, 368, 56, 1623, 2649, 753, 15, 10176, 20544, 9024, 576, 71793, 172596, 104814, 13572, 105, 562848, 1569408, 1228608, 259968, 7968, 4853949, 15398829, 14824314, 4532034, 306729, 945, 45664896, 162412416, 185991936, 75929856 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A permutation p(1)p(2)...p(n) in the symmetric group S_n contains the subword (132) if there are 3 consecutive elements p(i)p(j)p(k) that have the same order relations as (132), that is, p(i) < p(j) > p(k) and p(i) < p(k). For the enumeration of permutations containing the subword (123) see A162975. LINKS Alois P. Heinz, Rows n = 0..100, flattened S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125. FORMULA E.g.f.: 1/(1-int {t = 0..z} exp((u-1)*t^2/2!)) = sqrt(1-u)/(sqrt(1-u)-sqrt(Pi/2)*erf(z/2*sqrt(1-u))) = 1 + z + 2*z^2/2! +(5+u)*z^3/3! + (16+8*u)*z^4/4! + .... n-th row sum = n!. First column A111004. EXAMPLE Table begins .n\k.|......0......1.....2......3 = = = = = = = = = = = = = = = = = ..0..|......1 ..1..|......1 ..2..|......2 ..3..|......5......1 ..4..|.....16......8 ..5..|.....63.....54.....3 ..6..|....296....368....56 ..7..|...1623...2649...753....15 ..8..|..10176..20544..9024...576 ... T(4,0) = 16: The 16 permutations of S_4 not containing the subword (132) are (1234), (2134), (2314), (3124), (3214), (1342), (2341), (3241), (2413), (3412), (3421), (4123), (4213), (4231), (4312), (4321). T(4,1) = 8: The 8 permutations of S_4 with 1 occurrence of the subword (132) are 1243, 1324, 1423, 1432, 2143, 2431, 3142, 4132. MAPLE b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(       add(b(u-j, o+j-1, 0)*`if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0\$2)): seq(T(n), n=0..14);  # Alois P. Heinz, Oct 30 2013 MATHEMATICA b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, 0]*If[j

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Last modified January 15 19:35 EST 2019. Contains 319171 sequences. (Running on oeis4.)