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A333726
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Triangle read by rows: T(n,k) is the number of permutations in S_n for which all cycles are length k or greater.
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2
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1, 2, 1, 6, 2, 2, 24, 9, 6, 6, 120, 44, 24, 24, 24, 720, 265, 160, 120, 120, 120, 5040, 1854, 1140, 720, 720, 720, 720, 40320, 14833, 8988, 6300, 5040, 5040, 5040, 5040, 362880, 133496, 80864, 58464, 40320, 40320, 40320, 40320, 40320
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OFFSET
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1,2
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LINKS
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Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.
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FORMULA
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T(n,k) = 0 when n < k,
T(n,k) = (n-1)! when n = k, and otherwise
T(n,k) = (n-1) * T(n-1, k) + A068424(n-1, k-1) * T(n-k, k).
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EXAMPLE
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Triangle begins:
n\k| 1 2 3 4 5 6 7 8 9
---+--------------------------------------------------------
1| 1
2| 2 1
3| 6 2 2
4| 24 9 6 6
5| 120 44 24 24 24
6| 720 265 160 120 120 120
7| 5040 1854 1140 720 720 720 720
8| 40320 14833 8988 6300 5040 5040 5040 5040
9| 362880 133496 80864 58464 40320 40320 40320 40320 40320
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CROSSREFS
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Columns 1-4: A000142, A000166, A038205, A047865.
Cf. A068424.
Sequence in context: A351442 A325815 A292441 * A306549 A198870 A050457
Adjacent sequences: A333723 A333724 A333725 * A333727 A333728 A333729
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KEYWORD
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nonn,tabl
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AUTHOR
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Peter Kagey, Apr 23 2020
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STATUS
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approved
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