A393571 Triangle read by rows: T(n,k) is the number of 4-dimensional balanced ballot paths of length 4n with exactly k semisymmetric peaks. Here, 1 <= k <= 2n-1 if n >=1, and k=0 if n=0.

1, 1, 4, 9, 1, 25, 175, 216, 45, 1, 196, 2828, 9285, 9038, 2514, 162, 1, 1764, 43508, 274138, 613545, 533694, 176091, 19541, 522, 1, 17424, 658725, 6797510, 27289044, 48514811, 39765498, 14758731, 2299984, 126465, 1611, 1, 184041, 9925361, 152486372, 962836198, 2893816488, 4421263632, 3504993667, 1421502706, 280418027, 24232150, 742360, 4887, 1

Offset

0,3

Comments

A 4-dimensional balanced ballot path is a sequence of 4n steps starting at (0,0,0,0) and ending at (n,n,n,n), such that each step is a standard unit vector (i.e., is equal to either (1,0,0,0), (0,1,0,0), (0,0,1,0), or (0,0,0,1)) and each intermediate point of the path satisfies x_1 >= x_2 >= x_3 >= x_4.

A semisymmetric peak in a 4-dimensional balanced ballot path P is any step in P equal to (1,0,0,0) or (0,1,0,0) that is immediately followed by a step equal to either (0,0,1,0) or (0,0,0,1).

T(n, k) is defined as the number of 4-dimensional balanced ballot paths with 4n steps and exactly k semisymmetric peaks.

Links

Alois P. Heinz, Rows n = 0..60, flattened

Ryota Inagaki and Dimana Pramatarova, On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers, arXiv:2604.04900 [math.CO], 2026. See p. 32.

Example

Triangle T(n,k) begins:
     1;
     1;
     4,     9,      1;
    25,   175,    216,     45,      1;
   196,  2828,   9285,   9038,   2514,    162,     1;
  1764, 43508, 274138, 613545, 533694, 176091, 19541, 522, 1;
  ...

Prog

(Python)
from functools import lru_cache
@lru_cache(None)
def count_dyck_words(x_rem, y_rem, z_rem, w_rem, x_used, y_used, z_used, w_used, last_char, s_needed):
    if x_rem == y_rem == z_rem == w_rem == 0:
        return 1 if s_needed == 0 else 0
    if not (x_used >= y_used >= z_used >= w_used):
        return 0
    if s_needed < 0:
        return 0
    total = 0
    if x_rem > 0:
        total += count_dyck_words(x_rem - 1, y_rem, z_rem, w_rem, x_used + 1, y_used, z_used, w_used, 0, s_needed)
    if y_rem > 0 and x_used >= y_used + 1:
        total += count_dyck_words(x_rem, y_rem - 1, z_rem, w_rem, x_used, y_used + 1, z_used, w_used, 1, s_needed)
    if z_rem > 0 and y_used >= z_used + 1:
        used = 1 if (last_char == 0 or last_char==1)  else 0
        total += count_dyck_words(x_rem, y_rem, z_rem - 1, w_rem, x_used, y_used, z_used + 1, w_used, 2, s_needed - used)
    if w_rem > 0 and z_used >= w_used + 1:
        used = 1 if (last_char == 0 or last_char==1) else 0
        total += count_dyck_words(x_rem, y_rem, z_rem, w_rem - 1, x_used, y_used, z_used, w_used + 1, 3, s_needed - used)
    return total
print(count_dyck_words(0, 0, 0, 0, 0, 0, 0, 0, -1, 0))
for n in range(1, 7):
    for s in range(1, 2*n):
        result = count_dyck_words(n, n, n, n, 0, 0, 0, 0, -1, s)
        print(result)

Crossrefs

Cf. A393747, A393573, A393594, A387936.

Row sums are equal to A005790.

First column is equal to A001246.

Sequence in context: A152205 A129861 A055491 * A032523 A350623 A032760

Adjacent sequences: A393568 A393569 A393570 * A393572 A393573 A393574

Keyword

nonn,tabf

Author

Ryota Inagaki and Dimana Pramatarova, Feb 21 2026

Status

approved