A137782 a(n) = the number of permutations (p(1), p(2), ..., p(n)) of (1,2,...,n) where, for each k (2 <= k <= n), the sign of (p(k) - p(k-1)) equals the sign of (p(n+2-k) - p(n+1-k)).

1, 1, 2, 2, 12, 24, 200, 540, 6160, 21616, 306432, 1310880, 22338624, 113017696, 2245983168, 13108918368, 297761967360, 1969736890624, 50332737128960, 372125016868608, 10565549532009472, 86337114225206784, 2696451226217269248, 24132714802787013632

Offset

0,3

Comments

Number of order n permutations whose descent set is invariant w.r.t. the function f(x) = n-x. - Max Alekseyev, May 06 2009

Links

Alois P. Heinz, Table of n, a(n) for n = 0..190

Formula

a(2n) = A000984(n)*A060350(n). - Max Alekseyev, Apr 23 2009

Example

Consider the permutation (for n = 7): 3,6,7,5,1,2,4.
The signs of the differences between adjacent terms form the sequence: ++--++, which has reflective symmetry. So this permutation, among others, is counted when n = 7.

Maple

b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= proc(n) option remember; local r;
     `if`(irem(n, 2, 'r')=0, b(0, r$2)*binomial(n, r),
      add(add(binomial(j-1, i)*binomial(n-j, r-i)*
      b(r-i, i, n-j-r+i), i=0..min(j-1, r)), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2015

Mathematica

b[u_, o_, h_] := b[u, o, h] = If[u+o == 0, 1, Sum[Sum[b[u-j, o+j-1, h+i-1], {i, 1, u+o-h}], {j, 1, u}] + Sum[Sum[b[u+j-1, o-j, h-i], {i, 1, h}], {j, 1, o}]];
a[n_] := a[n] = Module[{r = Quotient[n, 2]}, If[Mod[n, 2] == 0, b[0, r, r]*Binomial[n, r], Sum[Sum[Binomial[j-1, i]*Binomial[n-j, r-i]*b[r-i, i, n-j-r+i], {i, 0, Min[j-1, r]}], {j, 1, n}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 13 2019, after Alois P. Heinz *)

Prog

(PARI) { a(n) = local(r, u, c, t); r=0; forvec(v=vector(n-1, i, [2*i==n, 1]), u=sum(i=1, #v, v[i]); c=sum(i=1, (n-1)\2, !v[i]&&!v[n-i]); t=[0]; for(i=1, #v, if(v[i], t=concat(t, [i]))); r += (-1)^u * 2^c * n! \ prod(i=2, #t, (t[i]-t[i-1])!) \ (n-t[ #t])! ); (-1)^(n+1)*r } \\ Max Alekseyev, May 06 2009

Crossrefs

Cf. A137783.

Sequence in context: A362192 A122007 A365643 * A131384 A052612 A287951

Adjacent sequences: A137779 A137780 A137781 * A137783 A137784 A137785

Keyword

nonn

Author

Leroy Quet, Feb 10 2008

Extensions

First 8 terms calculated by Olivier Gérard

Extended by Max Alekseyev, May 06 2009

a(0), a(22) from Alois P. Heinz, Jul 02 2015

a(23) from Alois P. Heinz, Sep 15 2015

Status

approved