OFFSET
1,3
COMMENTS
For point x (x_1,x_2,x_3,x_4) in the 4-dimensional lattice, we define the height of x as h_4(x) := 3x_1 - x_2 - x_3 - x_4. The 4-dimensional balanced ballot path (multidimensional Dyck path), is a sequence of 4*n steps with initial point (0,0,0,0) and ending at (n,n,n,n) satisfying that each step is a standart unit vector and each point of the path satisfies x_1 >= x_2 >= x_3 >= x_4. T(n,k) is the number of 4-dimensional balanced ballot paths of 4*n steps such that the largest height reached by any point in the path is equal to k, i.e. for at least one intermediate point h(x) = k, but for no points h(x) > k.
LINKS
Sean A. Irvine, Table of n, a(n) for n = 1..3725 (rows 1..50 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. 27.
EXAMPLE
Triangle begins:
1;
1, 3, 5, 5;
1, 15, 68, 147, 105, 84, 42;
1, 63, 722, 3098, 4720, 5940, 5112, 2520, 1386, 462;
1, 255, 7100, 58130, 157631, 297458, 374568, 306783, 236544, 137676, 56628, 24024, 6006;
...
PROG
(Python)
"""change values for the endpoint t(n, n, n, n) and the height bd=k"""
t = (5, 5, 5, 5)
bd = 5
def ht(t):
return 3 * t[0] - t[1] - t[2] - t[3]
def altern(t, bd):
count = 0
if t == (0, 0, 0, 0):
return 1
if t[0] > 0 and t[0] - 1 >= t[1]:
count += altern((t[0] - 1, t[1], t[2], t[3]), bd)
if t[1] > 0 and t[1] - 1 >= t[2] and ht((t[0], t[1] - 1, t[2], t[3])) <= bd:
count += altern((t[0], t[1] - 1, t[2], t[3]), bd)
if t[2] > 0 and t[2] - 1 >= t[3] and ht((t[0], t[1], t[2] - 1, t[3])) <= bd:
count += altern((t[0], t[1], t[2] - 1, t[3]), bd)
if t[3] > 0 and ht((t[0], t[1], t[2], t[3] - 1)) <= bd:
count += altern((t[0], t[1], t[2], t[3] -1), bd)
return count
def exact_height(t, bd):
if bd < 0:
return 0
return altern(t, bd) - altern(t, bd - 1)
print(exact_height(t, bd))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Dimana Pramatarova and Ryota Inagaki, Oct 13 2025
STATUS
approved