login
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.
3
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
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.
Sequence in context: A088140 A130758 A130892 * A239871 A147389 A147008
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 21 2022
STATUS
approved