A304001 - OEIS

%I #19 Mar 26 2021 12:36:43

%S 1,1,1,5,12,93,360,3728,20160,259535,1814400,27820524,239500800,

%T 4251096402,43589145600,877606592736,10461394944000,235288904377275,

%U 3201186852864000,79476406782222500,1216451004088320000,33020655481590446318,562000363888803840000

%N Number of permutations of [n] whose up-down signature has a nonnegative total sum.

%C The up-down signature has (+1) for each ascent and (-1) for each descent.

%H Alois P. Heinz, <a href="/A304001/b304001.txt">Table of n, a(n) for n = 0..450</a>

%p b:= proc(u, o, t) option remember; (n->

%p      `if`(t>=n, n!, `if`(t<-n, 0,

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

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

%p     end:

%p a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 0), j=1..n)):

%p seq(a(n), n=0..25);

%p # second Maple program:

%p a:= n-> `if`(irem(n, 2, 'r')=0, ceil(n!/2),

%p          add(combinat[eulerian1](n, j), j=0..r)):

%p seq(a(n), n=0..25);

%t Eulerian1[n_, k_] := If[k == 0, 1, If[n == 0, 0, Sum[(-1)^j (k - j + 1)^n Binomial[n + 1, j], {j, 0, k + 1}]]];

%t a[n_] := Module[{r, m}, {r, m} = QuotientRemainder[n, 2]; If[m == 0, Ceiling[n!/2], Sum[Eulerian1[n, j], {j, 0, r}]]];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Mar 26 2021, after 2nd Maple program *)

%Y Bisections give: A002674 (even part), A179457(2n+1,n+1) (odd part).

%Y Cf. A000246 (for nonnegative partial sums), A006551 (total sums are 0 or 1), A008292, A303287.

%K nonn

%O 0,4

%A _Alois P. Heinz_, May 04 2018