A390883 - OEIS

%I #28 Dec 07 2025 15:23:53

%S 1,1,1,2,1,10,1,37,10,1,126,105,1,422,714,70,1,1422,4032,1176,1,4853,

%T 20628,11928,588,1,16786,99495,95040,13860,1,58775,461945,656205,

%U 189090,5544,1,208000,2089230,4121260,1958385,169884,1,742887,9271470,24202750,17090645,2930499,56628

%N 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).

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

%H Andrew Howroyd, <a href="/A390883/b390883.txt">Table of n, a(n) for n = 0..2600</a> (rows 0..100)

%H Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, <a href="https://arxiv.org/abs/2511.04815">Parking trees and the toric g-vector of nestohedra</a>, arXiv:2511.04815 [math.CO], 2025.

%F 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).

%F 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).

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

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

%e 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.

%e Triangle begins:

%e   1;

%e   1;

%e   1, 2;

%e   1, 10;

%e   1, 37, 10;

%e   1, 126, 105;

%e   1, 422, 714, 70;

%e   1, 1422, 4032, 1176;

%e   1, 4853, 20628, 11928, 588;

%e   1, 16786, 99495, 95040, 13860;

%e   1, 58775, 461945, 656205, 189090, 5544;

%e   ...

%p 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

%o (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

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

%K nonn,tabf

%O 0,4

%A _Richard Ehrenborg_, Nov 22 2025