A326412 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, 5, 17, 69, 330, 1797, 11028, 74932, 559351, 4540088, 39840318, 375421225, 3782383945, 40548234374, 460956742449, 5536790753853, 70077462043662, 931945968071778, 12993337101354500, 189485727877247991, 2884989393948284323, 45772604755492432599
Offset: 0
Keywords
Examples
a(4) = 17: 0000, 0001, 0002, 0003, 0010, 0011, 0020, 0021, 0022, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..483
- Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.
Crossrefs
Programs
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Maple
b:= proc(n, j, t, u, c) option remember; `if`(n=0, 1, add( `if`(c>0 or i>=j and t or i<=j and u, b(n-1, i, is(i<=j), is(i>=j), max(0, c-1)), 0), i=1..n)) end: a:= n-> b(n, 0, true$2, 2): seq(a(n), n=0..25); -
Mathematica
b[n_, j_, t_, u_, c_] := b[n, j, t, u, c] = If[n == 0, 1, Sum[If[c > 0 || i >= j && t || i <= j && u, b[n - 1, i, i <= j, i >= j , Max[0, c - 1]], 0], {i, 1, n}]]; a[n_] := b[n, 0, True, True, 2]; a /@ Range[0, 25] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)
Formula
a(n) ~ n! * c * 2^n * n^((Pi+1)/2) / Pi^n, where c = 0.0662002484840446134... - Vaclav Kotesovec, Oct 31 2019