A391402 - OEIS

%I #28 Dec 28 2025 18:47:18

%S 1,1,1,3,1,16,1,65,20,1,246,225,1,917,1659,175,1,3424,10192,3136,1,

%T 12861,56664,34104,1764,1,48610,296055,291600,44100,1,184745,1482965,

%U 2157705,639540,19404,1,705420,7205550,14488540,7040385,627264,1,2704143,34213530,90714910,65224445,11424699,226512

%N 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).

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

%H Paolo Xausa, <a href="/A391402/b391402.txt">Table of n, a(n) for n = 0..10200</a> (rows 0..200 of triangle, flattened).

%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)*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. 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).

%F T(2*n,n) = A000891(n).

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

%e 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.

%e Triangle begins:

%e   1;

%e   1;

%e   1,      3;

%e   1,     16;

%e   1,     65,      20;

%e   1,    246,     225;

%e   1,    917,    1659,     175;

%e   1,   3424,   10192,    3136;

%e   1,  12861,   56664,   34104,   1764;

%e   1,  48610,  296055,  291600,  44100;

%e   1, 184745, 1482965, 2157705, 639540, 19404;

%e   ...

%p 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:

%p seq(seq(T(n,k), k=0..n/2), n=0..11);

%t 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]}];

%t Table[A391402[n, k], {n, 0, 15}, {k, 0, Quotient[n, 2]}] (* _Paolo Xausa_, Dec 28 2025 *)

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

%K nonn,tabf

%O 0,4

%A _Richard Ehrenborg_, Dec 08 2025

%E More terms from _Paolo Xausa_, Dec 28 2025