A328491 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j <= k or i <= j > k.

1, 1, 2, 1, 4, 6, 32, 89, 592, 2402, 19072, 101866, 939136, 6221228, 65291264, 516212409, 6075261184, 55812055946, 727912302592, 7618369901774, 109058247342080, 1280820543489044, 19965414947799040, 259988000952099210, 4383593333171363840, 62680335913868539796

Offset: 0

Author

Alois P. Heinz, Oct 16 2019

Keywords

• nonn

Examples

			a(0) = 1: the empty sequence.
a(1) = 1: 0.
a(2) = 2: 00, 01.
a(3) = 1: 010.
a(4) = 4: 0100, 0101, 0102, 0103.
a(5) = 6: 01010, 01020, 01021, 01030, 01031, 01032.
a(6) = 32: 010100, 010101, 010102, 010103, 010104, 010105, 010200, 010201, 010202, 010203, 010204, 010205, 010211, 010212, 010213, 010214, 010215, 010300, 010301, 010302, 010303, 010304, 010305, 010311, 010312, 010313, 010314, 010315, 010322, 010323, 010324, 010325.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..485
• Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Crossrefs

Cf. A000108, A000142, A000225, A001250, A328357, A328358, A328409, A328425.

Programs

Maple

b:= proc(n, j, t, c) option remember; `if`(n=0, 1, add(`if`(c=0 and
      (i>j xor t), 0, b(n-1, i, is(i<=j), max(0, c-1))), i=1..n))
    end:
a:= n-> b(n, 0, true, 2):
seq(a(n), n=0..27);

Mathematica

b[n_, j_, t_, c_] := b[n, j, t, c] = If[n == 0, 1, Sum[If[Xor[i > j, t] && c == 0, 0, b[n - 1, i, i <= j, Max[0, c - 1]]], {i, 1, n}]];
a[n_] := b[n, 0, True, 2];
a /@ Range[0, 27] (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)

Formula

a(n) is odd <=> n in { A000225 }.

a(n) ~ n! * c * 2^n / Pi^n, where c = 0.292816379603485707589209784583144390652038770449692132953726209770208058... - Vaclav Kotesovec, Oct 17 2019