A328409 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, 3, 6, 16, 57, 245, 1248, 7151, 46104, 325560, 2523437, 21106494, 190806861, 1842347541, 19018910502, 208088481921, 2414462433024, 29512737830802, 380156646308541, 5133381861786182, 72678441538790901, 1074324277172134786, 16581261996774703606

Offset

0,3

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.

Formula

a(n) ~ n! * c * 2^n * n^(Pi/4 - 1/2) / Pi^n, where c = 0.52096414784... - Vaclav Kotesovec, Oct 31 2019

Example

a(0) = 1: the empty sequence.
a(1) = 1: 0.
a(2) = 2: 00, 01.
a(3) = 3: 000, 010, 011.
a(4) = 6: 0000, 0101, 0102, 0103, 0110, 0111.
a(5) = 16: 00000, 01010, 01011, 01020, 01021, 01022, 01030, 01031, 01032, 01033, 01101, 01102, 01103, 01104, 01110, 01111.

Maple

b:= proc(n, j, t, c) option remember; `if`(n=0, 1, add(`if`((i>j
     xor t) and c=0, 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..24);

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, 24] (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000108, A328357, A328358, A328425, A328491.

Sequence in context: A305190 A087983 A340911 * A103593 A174118 A368947

Adjacent sequences: A328406 A328407 A328408 * A328410 A328411 A328412

Keyword

nonn

Author

Alois P. Heinz, Oct 14 2019

Status

approved