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A352523
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Number of integer compositions of n with exactly k nonfixed points (parts not on the diagonal).
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25
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1, 1, 0, 0, 2, 0, 1, 1, 2, 0, 0, 4, 2, 2, 0, 0, 5, 5, 4, 2, 0, 1, 3, 12, 8, 6, 2, 0, 0, 7, 14, 19, 14, 8, 2, 0, 0, 8, 21, 33, 32, 22, 10, 2, 0, 0, 9, 30, 54, 63, 54, 32, 12, 2, 0, 1, 6, 47, 80, 116, 116, 86, 44, 14, 2, 0, 0, 11, 53, 129, 194, 229, 202, 130, 58, 16, 2, 0
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OFFSET
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0,5
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COMMENTS
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A nonfixed point in a composition c is an index i such that c_i != i.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
0 2 0
1 1 2 0
0 4 2 2 0
0 5 5 4 2 0
1 3 12 8 6 2 0
0 7 14 19 14 8 2 0
0 8 21 33 32 22 10 2 0
0 9 30 54 63 54 32 12 2 0
1 6 47 80 116 116 86 44 14 2 0
For example, row n = 6 counts the following compositions (empty column indicated by dot):
(123) (6) (24) (231) (2112) (21111) .
(15) (33) (312) (2121) (111111)
(42) (51) (411) (3111)
(114) (1113) (11112)
(132) (1122) (11121)
(141) (1311) (11211)
(213) (2211)
(222) (12111)
(321)
(1131)
(1212)
(1221)
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MATHEMATICA
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pnq[y_]:=Length[Select[Range[Length[y]], #!=y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pnq[#]==k&]], {n, 0, 9}, {k, 0, n}]
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CROSSREFS
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The version for permutations is A098825.
The corresponding rank statistic is A352513.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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