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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 (list; graph; refs; listen; history; text; internal format)
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 forms 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

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 } [From Max Alekseyev, May 06 2009]

CROSSREFS

Cf. A137783.

Sequence in context: A092900 A164961 A122007 * A131384 A052612 A130306

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

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Last modified March 27 12:31 EDT 2017. Contains 284176 sequences.