A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.

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

Author

Gus Wiseman, May 15 2022

Keywords

• nonn

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

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.

Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352824, A352825, A352830, A352872.

Programs

Mathematica

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]==0&]],{n,0,30}]

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