A060701 Table by antidiagonals of Mahonian numbers T(n,k): permutations of n letters with k inversions.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 1, 5, 4, 1, 0, 0, 0, 6, 9, 5, 1, 0, 0, 0, 5, 15, 14, 6, 1, 0, 0, 0, 3, 20, 29, 20, 7, 1, 0, 0, 0, 1, 22, 49, 49, 27, 8, 1, 0, 0, 0, 0, 20, 71, 98, 76, 35, 9, 1, 0, 0, 0, 0, 15, 90, 169, 174, 111, 44, 10, 1, 0, 0, 0, 0, 9, 101, 259, 343, 285

Offset: 0

Author

Henry Bottomley, Apr 25 2001

Keywords

• nonn
• tabl

Examples

			1;
0,1;
0,1,1;
0,0,2,1;
0,0,2,3,1;
0,0,1,5,4,1;
0,0,0,6,9,5,1; ...
[1, 4, 2, 3], [1, 3, 4, 2], [2, 1, 4, 3], [2, 3, 1, 4], [3, 1, 2, 4] have 2 inversions so T(4, 2)=5.
		

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Corollary 1.3.10, p. 21.

Crossrefs

A008302 is the main entry for these numbers. Columns include A000012, A000027, A000096, A005286, A005287, A005288. Diagonals include A000707, A001892, A001893, A001894, A005283, A005284, A005285.

Programs

PARI

T(n,k)=polcoeff(prod(j=1,n-1,sum(i=0,j,x^i)),k)

Formula

T(n, k)=sum_{j=0..n}[T(n-1, k-j)].

Product (1+x+...+x^k), k=1..n-1 = Sum T(n, k)x^k, k=0..n(n-1)/2.

Extensions

Additional comments from Michael Somos, Jun 23 2002.