Xiaomei Chen - cp's OEIS frontend

User: Xiaomei Chen

Xiaomei Chen has authored 2 sequences.

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

Xiaomei Chen, Jan 05 2025

			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;
  ...

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.

Row sums give A257520.

Column 1 gives A005043.

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

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).

A379838 Triangle read by rows: T(n,k) is the total number of humps with height k in all Motzkin paths of order n, n >= 2 and 1 <= k <= n/2.

Original entry on oeis.org

1, 3, 8, 1, 20, 5, 50, 19, 1, 126, 63, 7, 322, 196, 34, 1, 834, 588, 138, 9, 2187, 1728, 507, 53, 1, 5797, 5016, 1749, 253, 11, 15510, 14454, 5786, 1067, 76, 1, 41834, 41470, 18590, 4147, 416, 13, 113633, 118690, 58487, 15223, 1976, 103, 1, 310571, 339274, 181181, 53599, 8528, 635, 15

Offset: 2

Xiaomei Chen, Jan 04 2025

			Triangle begins:
   [2]     1;
   [3]     3;
   [4]     8,    1;
   [5]    20,    5;
   [6]    50,   19,   1;
   [7]   126,   63,   7;
   [8]   322,  196,  34,  1;
   [9]   834,  588, 138,  9;
  [10]  2187, 1728, 507, 53, 1;
  ...

Xiaomei Chen, Table of n, a(n) for n = 2..401
Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], Dec 2024.
David He and Daniel Tubbenhauer, Tensor powers of representations of (diagram) monoids, arXiv:2508.04054 [math.RT], 2025. See p. 10.

Row lengths give A004526.
Row sums give A097861.
Column 1 gives A140662.
Cf. A064189.

def A379838_triangel(dim):
    M = matrix(ZZ, dim, dim)
    for n in (2..dim+1):
        for k in (1..math.floor(n/2)+1):
            for i in range(n-2*k+1):
                if ((n-i)%2)==0:
                    M[n-2,k-1]=M[n-2, k-1]+(4*k)/(n-i+2*k)*binomial(n,i)*binomial(n-i-1,(n-i)/2+k-1)
    return M

G.f.: Sum_{n>=2, k>=1} T(n,k) * x^n * y^k = x^2 * M^2(x) * y / ((1-x) * (1 - x^2 * M^2(x) * y)), where M(x) is the g.f. for A001006.
T(n,k) = Sum_{i=0..n-2*k, i==n (mod 2)} (4*k) / (n-i+2*k) * binomial(n,i) * binomial(n-i-1,(n-i)/2+k-1).
T(n,k) = Sum_{i=2k-1..n-1} A064189(i,2k-1).
T(n,k) + T(n,k+1) = A064189(n,2k).

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User: Xiaomei Chen

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.

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