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Search: id:A001100
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| A001100 |
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Triangle read by rows: T(n,k) = number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1. |
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9
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| 1, 0, 2, 0, 4, 2, 2, 10, 10, 2, 14, 40, 48, 16, 2, 90, 230, 256, 120, 22, 2, 646, 1580, 1670, 888, 226, 28, 2, 5242, 12434, 12846, 7198, 2198, 366, 34, 2, 47622, 110320, 112820, 64968, 22120, 4448, 540, 40, 2, 479306, 1090270, 1108612, 650644, 236968, 54304, 7900, 748
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Number of permutations of 12...n such that exactly k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
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REFERENCES
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F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
J. Riordan, A recurrence for permutations without rising or falling successions. Ann. Math. Statist. 36 (1965), 708-710.
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FORMULA
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Let S_{n, k} = number of permutations of 12...n with exactly k rising or falling successions. Let S[n](t) = Sum_{k >= 0} S_{n, k}*t^k. Then S[0] = 1; S[1] = 1; S[2] = 2*t; S[3] = 4*t+2*t^2; for n >= 4, S[n] = (n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4].
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EXAMPLE
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1; 0,2; 0,4,2; 2,10,10,2; 14,40,48,16,2; ...
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CROSSREFS
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Diagonals give A002464, A086852, A086853, A086854, A086955.
Triangle in A086856 multiplied by 2. Cf. A010028.
Adjacent sequences: A001097 A001098 A001099 this_sequence A001101 A001102 A001103
Sequence in context: A081880 A037035 A112824 this_sequence A136265 A066910 A094405
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KEYWORD
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tabl,nonn
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AUTHOR
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njas, Aug 19 2003
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