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

%I A177477 #31 Mar 08 2022 10:21:33
%S A177477 1,1,2,6,19,70,331,1863,11637,81110,635550,5495339,51590494,524043395,
%T A177477 5743546943,67478821537,844983073638,11240221721390,158365579448315,
%U A177477 2355375055596386,36870671943986643,606008531691619131,10435226671431973345,187860338952519968538
%N A177477 Number of permutations of 1..n avoiding adjacent step pattern up, down, up.
%C A177477 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.
%H A177477 Alois P. Heinz, <a href="/A177477/b177477.txt">Table of n, a(n) for n = 0..468</a>
%H A177477 Sergey Kitaev, <a href="https://doi.org/10.1016/j.dam.2006.09.011">Introduction to partially ordered patterns</a>, Discrete Applied Mathematics 155 (2007), 929-944.
%F A177477 a(n) ~ c * d^n * n!, where d = A245758 = 0.7827041801715217018447074977..., c = 2.035127405829990832658061124449458067... . - _Vaclav Kotesovec_, Aug 22 2014
%p A177477 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A177477        add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
%p A177477       `if`(t=3, 0, add(b(u+j-1, o-j, 2), j=1..o)))
%p A177477     end:
%p A177477 a:= n-> b(n, 0, 1):
%p A177477 seq(a(n), n=0..25);  # _Alois P. Heinz_, Mar 10 2020
%t A177477 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
%t A177477      Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}] +
%t A177477      If[t == 3, 0, Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
%t A177477 a[n_] := b[n, 0, 1];
%t A177477 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 08 2022, after _Alois P. Heinz_ *)
%Y A177477 Column k=0 of A227884.
%Y A177477 Column k=5 of A242784.
%Y A177477 Cf. A227883, A245758.
%K A177477 nonn
%O A177477 0,3
%A A177477 Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and _R. H. Hardin_, May 10 2010 (17 terms)
%E A177477 a(18)-a(23) from _Alois P. Heinz_, Oct 06 2013
%E A177477 a(0)=1 prepended by _Alois P. Heinz_, Mar 10 2020