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A110952
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Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0<k<n-1.
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1
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1, 3, 3, 6, 11, 6, 10, 26, 26, 10, 15, 50, 71, 50, 15, 21, 85, 155, 155, 85, 21, 28, 133, 295, 379, 295, 133, 28, 36, 196, 511, 799, 799, 511, 196, 36, 45, 276, 826, 1519, 1849, 1519, 826, 276, 45, 55, 375, 1266, 2674, 3829, 3829, 2674, 1266, 375, 55, 66, 495, 1860
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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COMMENTS
| Permutations of [n] with exactly 2 descents and the descents are adjacent. Adjusting for initial index: row sums are A045618; first diagonal is A000217, the triangular numbers; 2nd diagonal is A051925; and 3rd diagonal is A001701, generalized Stirling numbers.
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FORMULA
| T(n,k) = k*C(n,k+1) - C(n,k) + 1
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EXAMPLE
| Triangle (beginning with n=3, k=1) is:
1
3 3
6 11 6
10 26 26 10
15 50 71 50 15
e.g. n=5, k = 2, T(5,2) = 11 = permutations of [5] with first run 2 long and last run 5-2-1 = 2 long, namely {14325, 15324, 15423, 24315, 25314, 25413, 34215, 35214, 35412, 45213, 45312}
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CROSSREFS
| Cf. A045618, A000217, A051925, A001701, A112858.
Sequence in context: A049924 A049926 A169944 * A025250 A094305 A057963
Adjacent sequences: A110949 A110950 A110951 * A110953 A110954 A110955
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| David J. Scambler (dscambler(AT)bmm.com), Nov 22 2006
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