A352833 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k fixed points, k = 0, 1.

1, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 5, 8, 7, 12, 10, 16, 14, 23, 19, 30, 26, 42, 35, 54, 47, 73, 62, 94, 82, 124, 107, 158, 139, 206, 179, 260, 230, 334, 293, 420, 372, 532, 470, 664, 591, 835, 740, 1034, 924, 1288, 1148, 1588, 1422, 1962, 1756, 2404, 2161

Offset: 0

Author

Gus Wiseman, Apr 08 2022

Keywords

• nonn
• tabf

Comments

A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists, so all columns k > 1 are zeros.

Conjecture:

(1) This is A064428 interleaved with A001522.

(2) Reversing rows gives A300788, the strict version of A300787.

Examples

			Triangle begins:
  0: {1,0}
  1: {0,1}
  2: {1,1}
  3: {2,1}
  4: {3,2}
  5: {4,3}
  6: {6,5}
  7: {8,7}
  8: {12,10}
  9: {16,14}
For example, row n = 7 counts the following partitions:
  (7)       (52)
  (61)      (421)
  (511)     (322)
  (43)      (3211)
  (4111)    (2221)
  (331)     (22111)
  (31111)   (1111111)
  (211111)
		

Crossrefs

Row sums are A000041.

The version for permutations is A008290, for nonfixed points A098825.

The columns appear to be A064428 and A001522.

The version counting strong nonexcedances is A114088.

The version for compositions is A238349, rank statistic A352512.

The version for reversed partitions is A238352.

Reversing rows appears to give A300788, the strict case of A300787.

A000700 counts self-conjugate partitions, ranked by A088902.

A115720 and A115994 count partitions by their Durfee square.

A330644 counts non-self-conjugate partitions, ranked by A352486.

Cf. A000701, A219282, A257990, A350839, A352513, A352521-A352525.

Programs

Mathematica

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],pq[#]==k&]],{n,0,15},{k,0,1}]