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

%I A244006 #18 Nov 09 2024 19:14:55
%S A244006 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,
%T A244006 110,125,137,142,142,137,125,110,93,74,56,41,27,17,10,5,2,1,1,2,5,10,
%U A244006 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
%N A244006 Triangle read by rows, q-multinomial coefficient generalization of 3-dimensional lattice paths from the origin to (m,m,m).
%C A244006 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.).
%C A244006 (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.
%C A244006 (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.
%C A244006 The sequence is palindromic and unimodal by row.
%D A244006 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; page 85-86.
%H A244006 Graham H. Hawkes, <a href="/A244006/b244006.txt">Table of n, a(n) for n = 0..427</a>
%F A244006 G.f. for row m: (Product_{i=1..3*m} (1-q^i))/(Product_{j=1..m} (1-q^j))^3.
%e A244006 Triangle begins:
%e A244006   1;
%e A244006   1, 2, 2, 1;
%e A244006   1, 2, 5, 7, 11, 12, 14, 12, 11, 7, 5, 2, 1;
%Y A244006 Cf. Row sums A006480.
%K A244006 nonn,tabf
%O A244006 0,3
%A A244006 _Graham H. Hawkes_, Jun 17 2014