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Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of eight.
2

%I #5 Jul 01 2018 17:39:34

%S 1,8,1040,9468,507355,5313447,214378961,2571977379,92953037066,

%T 1265907917962,44038999833044,674142774632948,23379215615715958,

%U 398561935596289153,14037530250073013445,264291741199540446059,9551899031473405653870,197148463934806397523934

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

%H Alois P. Heinz, <a href="/A316395/b316395.txt">Table of n, a(n) for n = 8..451</a>

%F a(n) = A262170(n) - A262169(n).

%p b:= proc(u, o, c, k) option remember;

%p `if`(c<0 or c>k, 0, `if`(u+o=0, 1,

%p add(b(u-j, o-1+j, c+1, k), j=1..u)+

%p add(b(u+j-1, o-j, c-1, k), j=1..o)))

%p end:

%p a:= n-> b(n, 0$2, 8)-b(n, 0$2, 7):

%p seq(a(n), n=8..25);

%Y Column k=8 of A258829.

%Y Cf. A262169, A262170.

%K nonn

%O 8,2

%A _Alois P. Heinz_, Jul 01 2018