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A373951
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Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of n - k.
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33
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1, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 2, 1, 1, 0, 7, 4, 4, 0, 1, 0, 14, 5, 6, 5, 1, 1, 0, 23, 14, 10, 10, 6, 0, 1, 0, 39, 26, 29, 12, 14, 6, 1, 1, 0, 71, 46, 54, 40, 19, 16, 9, 0, 1, 0, 124, 92, 96, 82, 64, 22, 22, 8, 1, 1, 0, 214, 176, 204, 144, 137, 82, 30, 26, 10, 0, 1, 0
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OFFSET
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0,7
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
1 1 0
3 0 1 0
4 2 1 1 0
7 4 4 0 1 0
14 5 6 5 1 1 0
23 14 10 10 6 0 1 0
39 26 29 12 14 6 1 1 0
71 46 54 40 19 16 9 0 1 0
124 92 96 82 64 22 22 8 1 1 0
Row n = 6 counts the following compositions:
(6) (411) (3111) (33) (222) (111111) .
(51) (114) (1113) (2211)
(15) (1311) (1221) (1122)
(42) (1131) (12111) (21111)
(24) (2112) (11211) (11112)
(141) (11121)
(321)
(312)
(231)
(213)
(132)
(123)
(2121)
(1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,2).
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MATHEMATICA
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Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-k&]], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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Column k = 0 is A003242 (anti-runs or compressed compositions).
Same as A373949 with rows reversed.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333382, A333489, A373952.
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KEYWORD
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AUTHOR
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STATUS
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approved
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