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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.
2
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
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).
Sequence in context: A349325 A134292 A344006 * A205123 A108715 A119671
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 21 2022
STATUS
approved