A303285 Number of permutations p of [2n] such that the sequence of ascents and descents of p0 forms a Dyck path.

1, 1, 8, 172, 7296, 518324, 55717312, 8460090160, 1726791794432, 456440969661508, 151770739970889792, 62022635037246022000, 30564038464166725328768, 17876875858414492985045712, 12245573879235563308351042496, 9711714975145772145881269175104

Offset

0,3

Comments

Here p is a permutation of 1,2,3,...,2n, and p0 refers to the string p followed by 0.

Also the number of permutations p of [2n] such that the sequence of ascents and descents of 0p forms a Dyck path. a(2) = 8: 1432, 2143, 2431, 3142, 3241, 3421, 4132, 4231.

Also the number of permutations p of [2n] that are of odd order and whose M statistic (as defined in the Spiro paper) is equal to n-1. - Sam Spiro, Nov 01 2018

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.

Wikipedia, Counting lattice paths

Formula

a(n) ~ c * 2^(2*n) * n^(2*n - 1) / exp(2*n), where c = 8.838022110416151362523442920999767406145711133564692... - Vaclav Kotesovec, May 22 2018

a(n) = (1/2)*Sum_{k odd} binomial(2*n,k)*A177042((k-1)/2)*A177042((2n-k-1)/2) for n>0. - Sam Spiro, Nov 01 2018

a(n) = A321280(2n,n-1) for n >= 1. - Alois P. Heinz, Nov 02 2018

Example

a(0) = 1: the empty permutation.
a(1) = 1: 12.
a(2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(0, 2*n, 0):
seq(a(n), n=0..20);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[0, 2n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from Maple *)

Prog

(PARI) \\ here b(n) is A177042
b(n)={if(n==0, 1, 2*sum(k=0, n, (-1)^k*binomial(2*n+1, k)*(n-k+1)^(2*n))); }
a(n)={if(n==0, 1, sum(k=1, n, binomial(2*n, 2*k-1)*b(k-1)*b(n-k))/2); } \\ Andrew Howroyd, Nov 01 2018

Crossrefs

Bisection (even part) of A303284.

Bisection (even part) of A303287.

Column k=0 of A316728.

Cf. A177042, A180056, A316727, A321280.

Sequence in context: A247729 A333460 A221060 * A061492 A263461 A215124

Adjacent sequences: A303282 A303283 A303284 * A303286 A303287 A303288

Keyword

nonn

Author

Alois P. Heinz, Apr 20 2018

Status

approved