A177477 Number of permutations of 1..n avoiding adjacent step pattern up, down, up.

1, 1, 2, 6, 19, 70, 331, 1863, 11637, 81110, 635550, 5495339, 51590494, 524043395, 5743546943, 67478821537, 844983073638, 11240221721390, 158365579448315, 2355375055596386, 36870671943986643, 606008531691619131, 10435226671431973345, 187860338952519968538

Offset: 0

Author

Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and R. H. Hardin, May 10 2010 (17 terms)

Keywords

• nonn

Comments

Suppose a < b, c < b, and c < d. To avoid abcd means not to have four consecutive letters such that the first letter is less than the second one, the third letter is less than the second one, and the third letter is less than the last one.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..468
• Sergey Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.

Crossrefs

Column k=0 of A227884.

Column k=5 of A242784.

Cf. A227883, A245758.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
       add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
      `if`(t=3, 0, add(b(u+j-1, o-j, 2), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2020

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}] +
     If[t == 3, 0, Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 08 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = A245758 = 0.7827041801715217018447074977..., c = 2.035127405829990832658061124449458067... . - Vaclav Kotesovec, Aug 22 2014

Extensions

a(18)-a(23) from Alois P. Heinz, Oct 06 2013

a(0)=1 prepended by Alois P. Heinz, Mar 10 2020