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Search: id:A060701
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| A060701 |
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Table by antidiagonals of Mahonian numbers T(n,k): permutations of n letters with k inversions. |
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+0
2
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| 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
(list; table; graph; listen)
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OFFSET
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0,9
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Corollary 1.3.10, p. 21.
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FORMULA
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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.
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EXAMPLE
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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.
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PROGRAM
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(PARI) T(n, k)=polcoeff(prod(j=1, n-1, sum(i=0, j, x^i)), k)
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CROSSREFS
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A008302 is the main entry for these numbers. Row sums are A000142. Columns include A000012, A000027, A000096, A005286, A005287, A005288. Diagonals include A000707, A001892, A001893, A001894, A005283, A005284, A005285.
Sequence in context: A120730 A122851 A064301 this_sequence A063181 A059220 A059431
Adjacent sequences: A060698 A060699 A060700 this_sequence A060702 A060703 A060704
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KEYWORD
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nonn,tabl
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Apr 25 2001
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EXTENSIONS
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Additional comments from Michael Somos, Jun 23, 2002.
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