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

0,9

References

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

Links

Table of n, a(n) for n=0..86.

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.

Example

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.

Prog

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

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.

Sequence in context: A122851 A064301 A199881 * A363916 A275345 A259668

Adjacent sequences: A060698 A060699 A060700 * A060702 A060703 A060704

Keyword

nonn,tabl

Author

Henry Bottomley, Apr 25 2001

Extensions

Additional comments from Michael Somos, Jun 23 2002.

Status

approved