|
| |
|
|
A244006
|
|
Triangle read by rows, q-multinomial coefficient generalization of 3-dimensional lattice paths from the origin to (m,m,m).
|
|
1
|
|
|
|
1, 1, 2, 2, 1, 1, 2, 5, 7, 11, 12, 14, 12, 11, 7, 5, 2, 1, 1, 2, 5, 10, 17, 27, 41, 56, 74, 93, 110, 125, 137, 142, 142, 137, 125, 110, 93, 74, 56, 41, 27, 17, 10, 5, 2, 1, 1, 2, 5, 10, 20, 33, 56, 86, 131, 186, 262, 350, 463, 586, 733, 885, 1056, 1219, 1391, 1542, 1689, 1799, 1894, 1942, 1968, 1942, 1894, 1799, 1689, 1542, 1391, 1219, 1056, 885, 733, 586, 463, 350, 262, 186, 131, 86, 56, 33, 20, 10, 5, 2, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
The sum of the elements in the m-th row gives the total number of lattice paths from the origin to (m,m,m), and the number of elements in the m-th row is 3m^2+1. The (m,n)-th entry gives the number of lattice paths from the origin to (m,m,m) with inversion number n. The inversion number of a given lattice path depends on the ordering of the coordinates, but the total number of lattice paths with inversion number n does not. To calculate the inversion number w.r.t. an ordering of coordinates, fix such an ordered set of coordinates (x_1, x_2, x_3) and represent a lattice path, L, as a sequence, S(L), of m copies of each of the numbers {1,2,3} in the natural way (a step in the x_1 direction corresponds to a 1, a step in the x_2 direction corresponds to a 2, etc.).
(1) There is a natural way to associate each sequence, S(L), to a set of two partitions, P_1 and P_2, where P_1 fits in an box of size m x m and P_2 fits in a box of size m x 2m. The inversion number of L is given by |P_1|+|P_2|. See link.
(2) Equivalently, the inversion number of L is given by summing over each entry in S(L), the number of entries which precede that entry and whose value exceeds that entry’s.
The sequence is palindromic and unimodal by row.
|
|
|
REFERENCES
|
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; page 85-86.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. (prod_{i=1}^{3m}(1-q^i))/((prod_{j=1}^{m}(1-q^j))^3.
|
|
|
EXAMPLE
|
1;
1, 2, 2, 1;
1, 2, 5, 7, 11, 12, 14, 12, 11, 7, 5, 2, 1;
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
