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A090806 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.

%I A090806 #25 Jul 18 2021 11:27:08
%S A090806 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 A090806 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 A090806 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 A090806 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 A090806 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 A090806 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 A090806 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 A090806 M. S. Cheema and T. S. Motzkin, "Multipartitions and multipermutations," Proc. Symp. Pure Math. 19 (1971), 39-70, eq. (3.1.3).
%H A090806 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 A090806 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 A090806 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A090806 G.f. for T(n, k): 1/Product_{i>=1, j>=1} (1 - w^i * z^j).
%F A090806 Recurrence: m*T(m, n) = Sum_{L>0, j>0, k>=0} j*T(m-L*j, n-L*k). [Cheema and Motzkin]
%F A090806 Also, Euler transform of the table whose g.f. is xy/((1-x)*(1-y)). - _Christian G. Bower_, Jun 03 2005
%e A090806 Triangle T(n,k) begins
%e A090806       1
%e A090806      1 1
%e A090806     1 2 1
%e A090806    1 2 2 1
%e A090806   1 3 4 3 1
%e A090806 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 A090806 Square array a(n,k) begins:
%e A090806   1 1 1  1  1 ...
%e A090806   1 2 2  3  3 ...
%e A090806   1 2 4  5  7 ...
%e A090806   1 3 5  9 12 ...
%e A090806   1 3 7 12 20 ...
%Y A090806 Cf. A108461. Main diagonal: A108469.
%K A090806 easy,nonn,tabl
%O A090806 2,5
%A A090806 _Don Knuth_, Feb 10 2004
%E A090806 More terms from _Christian G. Bower_, Jun 03 2005
%E A090806 Entry revised by _N. J. A. Sloane_, Jul 07 2005