A304001 Number of permutations of [n] whose up-down signature has a nonnegative total sum.

1, 1, 1, 5, 12, 93, 360, 3728, 20160, 259535, 1814400, 27820524, 239500800, 4251096402, 43589145600, 877606592736, 10461394944000, 235288904377275, 3201186852864000, 79476406782222500, 1216451004088320000, 33020655481590446318, 562000363888803840000

Offset: 0

Author

Alois P. Heinz, May 04 2018

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

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

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

Programs

Maple

b:= proc(u, o, t) option remember; (n->
     `if`(t>=n, n!, `if`(t<-n, 0,
      add(b(u-j, o+j-1, t-1), j=1..u)+
      add(b(u+j-1, o-j, t+1), j=1..o))))(u+o)
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 0), j=1..n)):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> `if`(irem(n, 2, 'r')=0, ceil(n!/2),
         add(combinat[eulerian1](n, j), j=0..r)):
seq(a(n), n=0..25);

Mathematica

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}]]];
a[n_] := Module[{r, m}, {r, m} = QuotientRemainder[n, 2]; If[m == 0, Ceiling[n!/2], Sum[Eulerian1[n, j], {j, 0, r}]]];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 26 2021, after 2nd Maple program *)