

A090806


Triangular array read by rows: T(n,k) (n >= 2, 1 <= k <= n) = number of partitions of k white balls and nk black balls in which each part has at least one ball of each color. Also limit of the joint majorindex / inversion polynomial for permutations of n elements, as n becomes infinite.


4



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
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OFFSET

2,5


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 GarsiaGessel reference.


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. 209228. 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), 3970, eq. (3.1.3).


LINKS

Table of n, a(n) for n=2..87.
A. M. Garsia and I. Gessel, Permutation statistics and partitions, Advances in Mathematics, Volume 31, Issue 3, March 1979, Pages 288305.
Günter Meinardus, Zur additiven Zahlentheorie in mehreren Dimensionen, Teil I, Math. Ann. 132 (1956), 333346. [Gives asymptotic growth]
N. J. A. Sloane, Transforms


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(mL*j, nL*k). [Cheema and Motzkin]
Also, Euler transform of the table whose g.f. is xy/((1x)*(1y)).  Christian G. Bower, Jun 03 2005


EXAMPLE

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; bbwwbw; bwbwbw; bbwbww.
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 ...


CROSSREFS

Cf. A108461. Main diagonal: A108469.
Sequence in context: A180980 A275298 A048570 * A241926 A174446 A071201
Adjacent sequences: A090803 A090804 A090805 * A090807 A090808 A090809


KEYWORD

easy,nonn,tabl


AUTHOR

Don Knuth, Feb 10 2004


EXTENSIONS

More terms from Christian G. Bower, Jun 03 2005
Entry revised by N. J. A. Sloane, Jul 07 2005


STATUS

approved



