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 A295974 Number of length-n permutations avoiding descent patterns aba and bab. 3
 1, 1, 2, 6, 14, 52, 204, 1010, 5466, 34090, 233026, 1765836, 14534404, 129916550, 1248875862, 12872804422, 141470905326, 1652327596652, 20430973234388, 266683791698634, 3664052636652962, 52859944626536554, 798893924217099426, 12622926284124944660 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The descent pattern of a permutation is the sign of the first difference of the permutation, with "a" denoting a rise (+1) and "b" denoting a descent (-1).  For example, the permutation 364125 has descent pattern abbaa. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..479 Richard Ehrenborg and JiYoon Jung, Descent pattern avoidance, Adv. Appl. Math. 49 (3-5) (2012), 375-390. Richard Ehrenborg and JiYoon Jung, Descent pattern avoidance, arXiv preprint:1312.2027 [math.CO], 6 Dec 2013. FORMULA Ehrenborg and Jung prove that a(n) ~ 0.8908970548...*(0.6869765032...)^(n-3)*n!. EXAMPLE For n = 4 the 10 permutations NOT counted are 1324, 1423, 2143, 2314, 2413, 3142, 3241, 3412, 4132, 4231. MAPLE b:= proc(u, o, t, h) option remember;            `if`(u+o=0, 1, `if`(t=0, add(b(u-j, j-1, 1\$2), j=1..u),        add(`if`(h=3, 0, b(u-j, o+j-1, [1, 3, 1][t], 2)), j=1..u)+        add(`if`(t=3, 0, b(u+j-1, o-j, 2, [1, 3, 1][h])), j=1..o)))     end: a:= n-> b(n, 0\$3): seq(a(n), n=0..30);  # Alois P. Heinz, Dec 01 2017 MATHEMATICA b[u_, o_, t_, h_] := b[u, o, t, h] = If[u + o == 0, 1, If[t == 0, Sum[b[u - j, j - 1, 1, 1], {j, 1, u}], Sum[If[h == 3, 0, b[u - j, o + j - 1, {1, 3, 1}[[t]], 2]], {j, 1, u}] + Sum[If[t == 3, 0, b[u + j - 1, o - j, 2, {1, 3, 1}[[h]]]], {j, 1, o}]]]; a[n_] := b[n, 0, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 28 2017, after Alois P. Heinz *) CROSSREFS Column k=0 of A295987. Cf. A296054. Sequence in context: A284701 A011455 A188491 * A324365 A192764 A055691 Adjacent sequences:  A295971 A295972 A295973 * A295975 A295976 A295977 KEYWORD nonn AUTHOR Jeffrey Shallit, Dec 01 2017 EXTENSIONS a(13)-a(23) from Alois P. Heinz, Dec 01 2017 STATUS approved

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Last modified March 19 23:02 EDT 2019. Contains 321343 sequences. (Running on oeis4.)