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%I #25 Jul 18 2021 11:27:08
%S 1,1,1,1,2,1,1,2,2,1,1,3,4,3,1,1,3,5,5,3,1,1,4,7,9,7,4,1,1,4,9,12,12,
%T 9,4,1,1,5,11,17,20,17,11,5,1,1,5,13,22,28,28,22,13,5,1,1,6,16,29,40,
%U 45,40,29,16,6,1,1,6,18,35,53,64,64,53,35,18,6,1,1,7,21,44,70,91,100,91
%N Triangular array read by rows: T(n,k) (n >= 2, 1 <= k <= n) = number of partitions of k white balls and n-k black balls in which each part has at least one ball of each color. Also limit of the joint major-index / inversion polynomial for permutations of n elements, as n becomes infinite.
%C Alternatively, square array read by antidiagonals: a(n,k) (n >= 1, k >= 1) = number of partitions of (n,k) into pairs (i,j) with i,j>0. The addition rule is (a,b)+(x,y)=(a+x,b+y). E.g., (4,3) = (3,2)+(1,1) = (3,1)+(1,2) = (2,2)+(2,1) = (2,1)+(1,1)+(1,1), so T(4,3)=5. - _Christian G. Bower_, Jun 03 2005
%C Permutations of n elements have a polynomial sum x^{ind pi}y^{inv pi} where ind denotes the major index and inv the number of inversions. For example when n=3 the polynomial is 1 + xy + xy^2 + x^2y + x^2y^2 + x^3y^3. The coefficient of x^i y^j when i+j <= n is given by this sequence; in other words, the polynomials approach 1 + xy + x^2y + xy^2 + x^3y + 2x^2y^2 + xy^3 + ... + 4x^3y^3 + ... as n grows. The reasons can be found in the Garsia-Gessel reference.
%D Alter, Ronald; Curtz, Thaddeus B.; Wang, Chung C. Permutations with fixed index and number of inversions. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 209-228. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. From _N. J. A. Sloane_, Mar 20 2012
%D M. S. Cheema and T. S. Motzkin, "Multipartitions and multipermutations," Proc. Symp. Pure Math. 19 (1971), 39-70, eq. (3.1.3).
%H A. M. Garsia and I. Gessel, <a href="https://doi.org/10.1016/0001-8708(79)90046-X">Permutation statistics and partitions</a>, Advances in Mathematics, Volume 31, Issue 3, March 1979, Pages 288-305.
%H Günter Meinardus, <a href="https://eudml.org/doc/160524">Zur additiven Zahlentheorie in mehreren Dimensionen, Teil I</a>, Math. Ann. 132 (1956), 333-346. [Gives asymptotic growth]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f. for T(n, k): 1/Product_{i>=1, j>=1} (1 - w^i * z^j).
%F Recurrence: m*T(m, n) = Sum_{L>0, j>0, k>=0} j*T(m-L*j, n-L*k). [Cheema and Motzkin]
%F Also, Euler transform of the table whose g.f. is xy/((1-x)*(1-y)). - _Christian G. Bower_, Jun 03 2005
%e Triangle T(n,k) begins
%e 1
%e 1 1
%e 1 2 1
%e 1 2 2 1
%e 1 3 4 3 1
%e The first row is for n=2. When n=6 and there are 3 balls of each color, the four partitions in question are bbbwww; bbww|bw; bw|bw|bw; bbw|bww.
%e Square array a(n,k) begins:
%e 1 1 1 1 1 ...
%e 1 2 2 3 3 ...
%e 1 2 4 5 7 ...
%e 1 3 5 9 12 ...
%e 1 3 7 12 20 ...
%Y Cf. A108461. Main diagonal: A108469.
%K easy,nonn,tabl
%O 2,5
%A _Don Knuth_, Feb 10 2004
%E More terms from _Christian G. Bower_, Jun 03 2005
%E Entry revised by _N. J. A. Sloane_, Jul 07 2005
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