A391402 Triangle read by rows: T(n,k) is k-th entry of the toric g-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).

1, 1, 1, 3, 1, 16, 1, 65, 20, 1, 246, 225, 1, 917, 1659, 175, 1, 3424, 10192, 3136, 1, 12861, 56664, 34104, 1764, 1, 48610, 296055, 291600, 44100, 1, 184745, 1482965, 2157705, 639540, 19404, 1, 705420, 7205550, 14488540, 7040385, 627264, 1, 2704143, 34213530, 90714910, 65224445, 11424699, 226512

Offset

0,4

Comments

Also the number of 123-avoiding functions f:[n]->[n] having exactly k ascents.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10200 (rows 0..200 of triangle, flattened).

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)*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. for row n: Sum_{j=0..floor(n/2)} binomial(2*j,j)*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) = A000891(n).

T(2*n+1,n) = A337900(n+1).

Example

T(3,1) = 16 since there are 16 functions f:[3]->[3] that are 123-avoiding and have exactly 1 ascent: 121, 131, 132, 211, 212, 213, 221, 231, 232, 311, 312, 313, 322, 323, 311, 332.
Triangle begins:
  1;
  1;
  1,      3;
  1,     16;
  1,     65,      20;
  1,    246,     225;
  1,    917,    1659,     175;
  1,   3424,   10192,    3136;
  1,  12861,   56664,   34104,   1764;
  1,  48610,  296055,  291600,  44100;
  1, 184745, 1482965, 2157705, 639540, 19404;
  ...

Maple

T:= 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(n/2), n-j)), j=0..floor(n/2)) end proc:
seq(seq(T(n, k), k=0..n/2), n=0..11);

Mathematica

A391402[n_, k_] := Sum[Binomial[2*j, j]*Binomial[n, 2*j]*Binomial[2*#, #]*Binomial[n - i, i]*(-1)^(i - k)*Binomial[i, k]/(# + 1) & [n - i - j], {j, 0, Quotient[n, 2]}, {i, 0, Min[Quotient[n, 2], n - j]}];
Table[A391402[n, k], {n, 0, 15}, {k, 0, Quotient[n, 2]}] (* Paolo Xausa, Dec 28 2025 *)

Crossrefs

Cf. A391403, A000891, A337900, A381676 (row sums).

Sequence in context: A160616 A362847 A168319 * A326374 A143565 A143018

Adjacent sequences: A391399 A391400 A391401 * A391403 A391404 A391405

Keyword

nonn,tabf

Author

Richard Ehrenborg, Dec 08 2025

Extensions

More terms from Paolo Xausa, Dec 28 2025

Status

approved