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A352828
Number of strict integer partitions y of n with no fixed points y(i) = i.
17
1, 0, 1, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 26, 32, 38, 46, 56, 66, 78, 92, 106, 123, 142, 162, 186, 214, 244, 280, 322, 368, 422, 484, 552, 630, 718, 815, 924, 1046, 1180, 1330, 1498, 1682, 1888, 2118, 2372, 2656, 2972, 3322, 3712, 4146, 4626
OFFSET
0,4
LINKS
FORMULA
G.f.: Sum_{n>=0} q^(n*(3*n+1)/2)*Product_{k=1..n} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022
EXAMPLE
The a(0) = 1 through a(12) = 12 partitions (A-C = 10..12; empty column indicated by dot; 0 is the empty partition):
0 . 2 3 4 5 6 7 8 9 A B C
21 31 41 51 43 53 54 64 65 75
61 71 63 73 74 84
431 81 91 83 93
432 532 A1 B1
531 541 542 642
631 632 651
4321 641 732
731 741
5321 831
5421
6321
MATHEMATICA
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&pq[#]==0&]], {n, 0, 30}]
CROSSREFS
The version for permutations is A000166, complement A002467.
The reverse version is A025147, complement A238395, non-strict A238394.
The non-strict version is A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A238351, complement A352875.
The complement is A352829, non-strict A001522 (unproved, ranked by A352827 or A352874).
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352833 counts partitions by fixed points.
Sequence in context: A035635 A114869 A316899 * A029100 A356758 A098133
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2022
STATUS
approved