A353319 Irregular triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k excedances (parts above the diagonal), all zeros removed.

1, 1, 1, 2, 1, 2, 3, 4, 2, 1, 5, 4, 2, 7, 6, 2, 10, 6, 6, 15, 7, 7, 1, 18, 14, 7, 3, 26, 15, 11, 4, 35, 17, 19, 6, 47, 24, 19, 11, 61, 33, 22, 18, 1, 80, 44, 28, 20, 4, 103, 54, 42, 25, 7, 138, 60, 57, 31, 11, 175, 85, 58, 52, 15, 224, 112, 66, 64, 24

Offset

1,4

Links

Table of n, a(n) for n=1..69.

Example

Triangle begins:
   1
   1  1
   2  1
   2  3
   4  2  1
   5  4  2
   7  6  2
  10  6  6
  15  7  7  1
  18 14  7  3
  26 15 11  4
  35 17 19  6
  47 24 19 11
  61 33 22 18  1
  80 44 28 20  4
For example, row n = 9 counts the following reversed partitions:
  (1134)       (9)     (27)   (234)
  (1224)       (18)    (36)
  (1233)       (117)   (45)
  (11115)      (126)   (135)
  (11124)      (1116)  (144)
  (11133)      (1125)  (225)
  (11223)      (2223)  (333)
  (12222)
  (111114)
  (111123)
  (111222)
  (1111113)
  (1111122)
  (11111112)
  (111111111)

Mathematica

partsabove[y_]:=Length[Select[Range[Length[y]], #<y[[#]]&]];
DeleteCases[Table[Length[Select[Reverse/@IntegerPartitions[n], partsabove[#]==k&]], {n, 1, 30}, {k, 0, n-1}], 0, 2]

Crossrefs

Row sums are A000041.

Row lengths are A003056.

The version for permutations is A008292, opposite A123125.

The weak unreversed version is A115720/A115994, rank statistic A257990.

For fixed points instead of excedances we have A238352, rank stat A352822.

Column k = 0 is A238875.

The version for compositions is A352524, weak A352525.

The version for unreversed partitions is A353318.

A000700 counts self-conjugate partitions, ranked by A088902.

A001522 counts partitions with a fixed point, ranked by A352827 (unproved).

A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).

Cf. A000701, A002620, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

Sequence in context: A349325 A134292 A344006 * A205123 A108715 A119671

Adjacent sequences: A353316 A353317 A353318 * A353320 A353321 A353322

Keyword

nonn,tabf

Author

Gus Wiseman, May 21 2022

Status

approved