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A333755
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Triangle read by rows where T(n,k) is the number of compositions of n with k runs, n >= 0, 0 <= k <= n.
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148
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1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 0, 2, 10, 4, 0, 0, 0, 4, 12, 14, 2, 0, 0, 0, 2, 22, 29, 10, 1, 0, 0, 0, 4, 26, 56, 36, 6, 0, 0, 0, 0, 3, 34, 100, 86, 31, 2, 0, 0, 0, 0, 4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0, 2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0
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OFFSET
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0,5
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COMMENTS
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Except for a(1) = 0, the data is identical to A238130 shifted right once. However, in A238130, each row after the first ends with a zero, while here each row after the first starts with a zero.
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 2 0
0 2 2 0
0 3 4 1 0
0 2 10 4 0 0
0 4 12 14 2 0 0
0 2 22 29 10 1 0 0
0 4 26 56 36 6 0 0 0
0 3 34 100 86 31 2 0 0 0
0 4 44 148 200 99 16 1 0 0 0
0 2 54 230 374 278 78 8 0 0 0 0
Row n = 6 counts the following compositions (empty column indicated by dot):
. (6) (15) (123) (1212)
(33) (24) (132) (2121)
(222) (42) (141)
(111111) (51) (213)
(114) (231)
(411) (312)
(1113) (321)
(1122) (1131)
(2211) (1221)
(3111) (1311)
(11112) (2112)
(21111) (11121)
(11211)
(12111)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[Split[#]]==k&]], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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The version for anti-runs is A106356.
The k-th composition in standard-order has A124767(k) runs.
The version counting descents is A238343.
The version counting weak ascents is A333213.
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KEYWORD
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AUTHOR
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STATUS
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approved
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