A244006 Triangle read by rows, q-multinomial coefficient generalization of 3-dimensional lattice paths from the origin to (m,m,m).

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

Offset: 0

Author

Graham H. Hawkes, Jun 17 2014

Keywords

• nonn
• tabf

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.

Examples

			Triangle begins:
  1;
  1, 2, 2, 1;
  1, 2, 5, 7, 11, 12, 14, 12, 11, 7, 5, 2, 1;
		

References

• D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; page 85-86.

Links

• Graham H. Hawkes, Table of n, a(n) for n = 0..427

Crossrefs

Cf. Row sums A006480.

Formula

G.f. for row m: (Product_{i=1..3*m} (1-q^i))/(Product_{j=1..m} (1-q^j))^3.