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A352875
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Number of integer compositions y of n with a fixed point y(i) = i.
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3
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0, 1, 1, 2, 5, 10, 21, 42, 86, 174, 351, 708, 1424, 2861, 5743, 11520, 23092, 46269, 92673, 185562, 371469, 743491, 1487870, 2977164, 5956616, 11916910, 23839736, 47688994, 95393322, 190811346, 381662507, 763389209, 1526881959, 3053930971, 6108131542, 12216698288
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OFFSET
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0,4
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LINKS
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FORMULA
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EXAMPLE
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The a(0) = 0 through a(5) = 10 compositions (empty column indicated by dot):
. (1) (11) (12) (13) (14)
(111) (22) (32)
(112) (113)
(121) (122)
(1111) (131)
(221)
(1112)
(1121)
(1211)
(11111)
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MATHEMATICA
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pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pq[#]>0&]], {n, 0, 15}]
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PROG
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(PARI)
S(v, u, c)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
seq(n)={my(v=vector(1+n), s=vector(#v, i, 2^(i-2))); v[1]=1; s[1]=0; for(i=1, n, v=S(v, vector(n, j, if(j==i, 'x, 1)), O(x)); s-=apply(p->polcoef(p, 0), v)); s} \\ Andrew Howroyd, Jan 02 2023
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CROSSREFS
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The complement for partitions is A064428, ranked by A352826 (unproved).
The complement is counted by A238351.
The case of just one fixed point is A240736.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352512 counts fixed points in standard compositions, nonfixed A352513.
A352833 counts partitions by fixed points.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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