A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 9, 1, 1, 12, 2, 1, 16, 5, 1, 20, 9, 1, 25, 16, 1, 30, 25, 1, 36, 39, 1, 1, 42, 56, 2, 1, 49, 80, 5, 1, 56, 109, 10, 1, 64, 147, 19, 1, 72, 192, 32, 1, 81, 249, 54, 1, 90, 315, 84, 1, 100, 396, 129, 1, 1, 110, 489, 190, 2, 1, 121, 600, 275, 5

Offset

1,5

Links

Table of n, a(n) for n=1..74.

Example

Triangle begins:
   1
   1   1
   1   2
   1   4
   1   6
   1   9   1
   1  12   2
   1  16   5
   1  20   9
   1  25  16
   1  30  25
   1  36  39   1
   1  42  56   2
   1  49  80   5
   1  56 109  10
For example, row n = 7 counts the following partitions:
  (1111111)  (7)       (43)
             (52)      (331)
             (61)
             (322)
             (421)
             (511)
             (2221)
             (3211)
             (4111)
             (22111)
             (31111)
             (211111)

Mathematica

partsabove[y_]:=Length[Select[Range[Length[y]], #<y[[#]]&]];
DeleteCases[Table[Length[Select[IntegerPartitions[n], partsabove[#]==k&]], {n, 1, 15}, {k, 0, n-1}], 0, 2]

Crossrefs

Row sums are A000041.

Row lengths are A000194, reversed A003056.

Column k = 1 is A002620, reversed A238875.

Column k = 2 is A097701.

The version for permutations is A008292, opposite A123125.

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

The version for compositions is A352524, weak A352525.

The version for reversed partitions is A353319.

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).

A238352 counts reversed partitions by fixed points, rank statistic A352822.

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

Sequence in context: A088140 A130758 A130892 * A239871 A147389 A147008

Adjacent sequences: A353315 A353316 A353317 * A353319 A353320 A353321

Keyword

nonn,tabf

Author

Gus Wiseman, May 21 2022

Status

approved