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A341439
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Table of generalized ménage numbers read by antidiagonals upward: T(n,k) is the number of permutations pi in S_k such that pi(i) != i, i+n (mod k) for all i; n, k >= 1.
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3
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0, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 2, 4, 13, 0, 0, 1, 2, 13, 80, 0, 1, 1, 9, 13, 82, 579, 0, 0, 2, 2, 13, 80, 579, 4738, 0, 1, 1, 4, 44, 82, 579, 4740, 43387, 0, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792, 0, 1, 2, 9, 13, 265, 579, 4752, 43390, 439794, 4890741
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OFFSET
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1,10
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COMMENTS
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The recurrence for the second row comes from Doron Zeilberger's MENAGE program, available via the arXiv reference.
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LINKS
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FORMULA
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T(n,k) = T(n+k, k).
T(2,k) = k*T(2,k-1) + 3*T(2,k-2) + (-2*k+6)*T(2,k-3) - 3*T(2,k-4) + (k-6)*T(2,k-5) + T(2,k-6) for k > 8.
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EXAMPLE
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Table begins:
n\k | 1 2 3 4 5 6 7 8
----+--------------------------
1 | 0 0 1 2 13 80 579 4738
2 | 0 1 1 4 13 82 579 4740
3 | 0 0 2 2 13 80 579 4738
4 | 0 1 1 9 13 82 579 4752
5 | 0 0 1 2 44 80 579 4738
6 | 0 1 2 4 13 265 579 4740
7 | 0 0 1 2 13 80 1854 4738
8 | 0 1 1 9 13 82 579 14833
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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