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.

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, 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, 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

Offset: 2

Author

Don Knuth, Feb 10 2004

Keywords

• easy
• nonn
• tabl

Comments

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

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.

Examples

			Triangle T(n,k) begins
      1
     1 1
    1 2 1
   1 2 2 1
  1 3 4 3 1
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.
Square array a(n,k) begins:
  1 1 1  1  1 ...
  1 2 2  3  3 ...
  1 2 4  5  7 ...
  1 3 5  9 12 ...
  1 3 7 12 20 ...
		

References

• 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
• M. S. Cheema and T. S. Motzkin, "Multipartitions and multipermutations," Proc. Symp. Pure Math. 19 (1971), 39-70, eq. (3.1.3).

Links

• A. M. Garsia and I. Gessel, Permutation statistics and partitions, Advances in Mathematics, Volume 31, Issue 3, March 1979, Pages 288-305.
• Günter Meinardus, Zur additiven Zahlentheorie in mehreren Dimensionen, Teil I, Math. Ann. 132 (1956), 333-346. [Gives asymptotic growth]
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A108461. Main diagonal: A108469.

Formula

G.f. for T(n, k): 1/Product_{i>=1, j>=1} (1 - w^i * z^j).

Recurrence: m*T(m, n) = Sum_{L>0, j>0, k>=0} j*T(m-L*j, n-L*k). [Cheema and Motzkin]

Also, Euler transform of the table whose g.f. is xy/((1-x)*(1-y)). - Christian G. Bower, Jun 03 2005

Extensions

More terms from Christian G. Bower, Jun 03 2005

Entry revised by N. J. A. Sloane, Jul 07 2005