A352832 Number of reversed integer partitions y of n with exactly one fixed point y(i) = i.

0, 1, 1, 1, 4, 3, 7, 7, 14, 19, 24, 32, 46, 60, 85, 109, 140, 179, 239, 300, 397, 495, 636, 790, 995, 1239, 1547, 1926, 2396, 2942, 3643, 4432, 5435, 6602, 8038, 9752, 11842, 14292, 17261, 20714, 24884, 29733, 35576, 42375, 50522, 60061, 71363, 84551, 100101

Offset: 0

Author

Gus Wiseman, Apr 08 2022

Keywords

• nonn

Comments

A reversed integer partition of n is a finite weakly increasing sequence of positive integers summing to n.

Examples

			The a(0) = 0 through a(8) = 14 partitions (empty column indicated by dot):
  .  (1)  (11)  (111)  (13)    (14)     (15)      (16)       (17)
                       (22)    (1112)   (114)     (115)      (116)
                       (112)   (11111)  (222)     (1123)     (134)
                       (1111)           (1113)    (11113)    (224)
                                        (1122)    (11122)    (233)
                                        (11112)   (111112)   (1115)
                                        (111111)  (1111111)  (2222)
                                                             (11114)
                                                             (11123)
                                                             (11222)
                                                             (111113)
                                                             (111122)
                                                             (1111112)
                                                             (11111111)
For example, the reversed partition (2,2,4) has a unique fixed point at the second position.
		

Crossrefs

* = unproved

*The non-reverse version is A001522, ranked by A352827, strict A352829.

*The non-reverse complement is A064428, ranked by A352826, strict A352828.

This is column k = 1 of A238352.

For no fixed point: counted by A238394, ranked by A352830, strict A025147.

For > 0 fixed points: counted by A238395, ranked by A352872, strict A096765.

The version for compositions is A240736, complement A352520.

These partitions are ranked by A352831.

A000700 counts self-conjugate partitions, ranked by A088902.

A008290 counts permutations by fixed points, nonfixed A098825.

A115720 and A115994 count partitions by their Durfee square.

A238349 counts compositions by fixed points, complement A352523.

A352822 counts fixed points of prime indices.

A352833 counts partitions by fixed points.

Cf. A064410, A177510, A257990, A325187, A351983, A352824, A352825.

Programs

Mathematica

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Reverse/@IntegerPartitions[n],pq[#]==1&]],{n,0,30}]