A390883 Triangle read by rows: T(n,k) is k-th entry of the toric g-vector of the n-dimensional associahedron, 0 <= k <= floor(n/2).

1, 1, 1, 2, 1, 10, 1, 37, 10, 1, 126, 105, 1, 422, 714, 70, 1, 1422, 4032, 1176, 1, 4853, 20628, 11928, 588, 1, 16786, 99495, 95040, 13860, 1, 58775, 461945, 656205, 189090, 5544, 1, 208000, 2089230, 4121260, 1958385, 169884, 1, 742887, 9271470, 24202750, 17090645, 2930499, 56628

Offset

0,4

Comments

Also the number of 123-avoiding parking functions of size n having exactly k ascents.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2600 (rows 0..100)

Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Parking trees and the toric g-vector of nestohedra, arXiv:2511.04815 [math.CO], 2025.

Formula

T(n,k) = Sum_{j=0..floor(n/2)} Sum_{i=0..min(floor(n/2),n-j)} (binomial(2*j, j)/(j+1)) * binomial(n,2*j) * binomial(2*n-2*i-2*j,n-i-j) * binomial(n-i,i) * (-1)^(i-k) * binomial(i,k)/(n-i-j+1).

G.f. of row n: Sum_{j=0..floor(n/2)} (binomial(2*j,j)/(j+1)) * binomial(n,2*j) * Sum_{k=0..min(floor(n/2),n-j)} binomial(2*(n-k-j),n-k-j) * binomial(n-k,k) * (x-1)^k / (n-k-j+1).

T(2*n,n) = A005568(n).

T(2*n+1,n) = A046715(n+1).

Example

T(4,2) = 10 since there are 10 parking functions of length 4 that are 123-avoiding and have exactly 2 ascents: 2211, 2311, 2312, 2411, 2412, 2413, 3311, 3312, 3411 and 3412.
Triangle begins:
  1;
  1;
  1, 2;
  1, 10;
  1, 37, 10;
  1, 126, 105;
  1, 422, 714, 70;
  1, 1422, 4032, 1176;
  1, 4853, 20628, 11928, 588;
  1, 16786, 99495, 95040, 13860;
  1, 58775, 461945, 656205, 189090, 5544;
  ...

Maple

proc(n, k) local j, i; add(binomial(2*j, j)*binomial(n, 2*j)*add(binomial(2*n-2*i-2*j, n-i-j)*binomial(n-i, i)*(-1)^(i-k)*binomial(i, k)/(n-i-j+1), i=0..min(floor(1/2*n), n-j))/(j+1), j=0..floor(1/2*n)); end proc

Prog

(PARI) T(n, k) = {sum(j=0, n\2, sum(i=0, min(n\2, n-j), (binomial(2*j, j)/(j+1)) * binomial(n, 2*j) * binomial(2*n-2*i-2*j, n-i -j) * binomial(n-i, i) * (-1)^(i-k) * binomial(i, k)/(n-i-j+1) ))} \\ Andrew Howroyd, Nov 27 2025

Crossrefs

Cf. A005568, A046715, A055151, A362741 (row sums).

Sequence in context: A055633 A105606 A132995 * A114692 A112691 A110169

Adjacent sequences: A390880 A390881 A390882 * A390884 A390885 A390886

Keyword

nonn,tabf

Author

Richard Ehrenborg, Nov 22 2025

Status

approved