A379893 Triangle read by rows: T(n,k) is the number of standard Young tableaux with shapes in {lambda = (lambda_1,lambda_2,...) | lambda_1-lambda_2=k, lambda_i<=1 for i>=3, |lambda| = n}, n >= 0 and 0 <= k <= n.

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 3, 3, 0, 1, 6, 9, 6, 4, 0, 1, 15, 21, 19, 10, 5, 0, 1, 36, 55, 50, 34, 15, 6, 0, 1, 91, 141, 139, 99, 55, 21, 7, 0, 1, 232, 371, 379, 293, 175, 83, 28, 8, 0, 1, 603, 982, 1043, 847, 551, 286, 119, 36, 9, 0, 1, 1585, 2628, 2872, 2441, 1684, 956, 441, 164, 45, 10, 0, 1

Offset: 0

Author

Xiaomei Chen, Jan 05 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  [0]   1;
  [1]   0,   1;
  [2]   1,   0,   1;
  [3]   1,   2,   0,  1;
  [4]   3,   3,   3,  0,  1;
  [5]   6,   9,   6,  4,  0,  1;
  [6]  15,  21,  19, 10,  5,  0, 1;
  [7]  36,  55,  50, 34, 15,  6, 0, 1;
  [8]  91, 141, 139, 99, 55, 21, 7, 0, 1;
  ...
		

Links

• Xiaomei Chen, Table of n, a(n) for n = 0..860
• Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], Dec 2024.

Crossrefs

Row sums give A257520.

Column 1 gives A005043.

Programs

Sage

def A379893_triangel(dim):
    M = matrix(ZZ, dim, dim)
    for n in range(dim):
        for k in range(n+1):
            for i in range(math.floor((n-k-1)/2)+1):
                for j in range(n-k-1-2*i+1):
                    if ((n+k-1-j)%2)==0:
                        M[n,k]=M[n, k]+(2*k+2)/(n+k+1-2*i-j)*binomial(n-2*i-2,j)*binomial(n-2*i-j-1,(n+k-j-1)/2-i)
            M[n,k]=M[n,k]-pow(-1,n+k+1)
    return M

Formula

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

T(n+1,2*k-1) + T(n,2*k-1) = A379838(n+1,k) - A379838(n,k).