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A316391 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of four.

%I A316391 #5 Jul 01 2018 13:43:23
%S A316391 1,4,62,340,4734,33855,495371,4403025,70083872,746704117,13023762276,
%T A316391 161905131484,3091115525637,43928623624790,914530883776894,
%U A316391 14623431780216366,330413968185491070,5870376151413374683,143271256595612492851,2799645366893284489691
%N A316391 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of four.
%H A316391 Alois P. Heinz, <a href="/A316391/b316391.txt">Table of n, a(n) for n = 4..455</a>
%F A316391 a(n) = A262166(n) - A262165(n).
%p A316391 b:= proc(u, o, c, k) option remember;
%p A316391       `if`(c<0 or c>k, 0, `if`(u+o=0, 1,
%p A316391        add(b(u-j, o-1+j, c+1, k), j=1..u)+
%p A316391        add(b(u+j-1, o-j, c-1, k), j=1..o)))
%p A316391     end:
%p A316391 a:= n-> b(n, 0$2, 4)-b(n, 0$2, 3):
%p A316391 seq(a(n), n=4..23);
%Y A316391 Column k=4 of A258829.
%Y A316391 Cf. A262165, A262166.
%K A316391 nonn
%O A316391 4,2
%A A316391 _Alois P. Heinz_, Jul 01 2018