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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)).
3
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
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
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