A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

0, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 26, 30, 36, 42, 50, 60, 70, 82, 96, 110, 126, 144, 163, 184, 208, 234, 264, 298, 336, 380, 430, 486, 550, 622, 702, 792, 892, 1002, 1125, 1260, 1408, 1572, 1752, 1950, 2168, 2408, 2672

Offset

0,7

Links

Table of n, a(n) for n=0..59.

Formula

G.f.: Sum_{n>=1} q^(n*(3*n-1)/2)*Product_{k=1..n-1} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

Example

The a(11) = 2 through a(17) = 12 partitions (A-F = 10..15):
  (92)   (A2)   (B2)    (C2)    (D2)     (E2)     (F2)
  (821)  (543)  (643)   (653)   (753)    (763)    (863)
         (921)  (A21)   (743)   (843)    (853)    (953)
                (5431)  (B21)   (C21)    (943)    (A43)
                        (5432)  (6432)   (D21)    (E21)
                        (6431)  (6531)   (6532)   (7532)
                                (7431)   (7432)   (7631)
                                (54321)  (7531)   (8432)
                                         (8431)   (8531)
                                         (64321)  (9431)
                                                  (65321)
                                                  (74321)

Mathematica

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

Crossrefs

The non-strict version is A001522 (unproved, ranked by A352827 or A352874).

The version for permutations is A002467, complement A000166.

The reverse version is A096765 (or A025147 shifted right once).

The non-strict reverse version is A238395, ranked by A352872.

The complement is counted by A352828, non-strict A064428 (unproved, ranked by A352826 or A352873).

The version for compositions is A352875, complement A238351.

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.

A238394 counts reversed partitions without a fixed point, ranked by A352830.

A352833 counts partitions by fixed points.

Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352823, A352824, A352825, A352832.

Sequence in context: A137722 A081305 A035665 * A307833 A331566 A358473

Adjacent sequences: A352826 A352827 A352828 * A352830 A352831 A352832

Keyword

nonn

Author

Gus Wiseman, May 15 2022

Status

approved