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A363126
Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-modes, all 0's removed.
7
1, 1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 8, 1, 10, 9, 3, 11, 13, 6, 15, 18, 9, 13, 24, 18, 1, 25, 24, 25, 3, 19, 36, 40, 6, 29, 41, 52, 13, 33, 45, 79, 19, 42, 57, 95, 36, 1, 39, 68, 133, 54, 3, 62, 72, 158, 87, 6, 55, 87, 214, 121, 13, 81, 95, 250, 177, 24
OFFSET
0,3
COMMENTS
A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.
EXAMPLE
Triangle begins:
1
1
2
3
4 1
4 3
8 3
6 8 1
10 9 3
11 13 6
15 18 9
13 24 18 1
25 24 25 3
19 36 40 6
29 41 52 13
33 45 79 19
42 57 95 36 1
39 68 133 54 3
Row n = 9 counts the following partitions:
(9) (441) (3321)
(54) (522) (4221)
(63) (711) (4311)
(72) (3222) (5211)
(81) (6111) (42111)
(333) (22221) (321111)
(432) (32211)
(531) (33111)
(621) (51111)
(222111) (411111)
(111111111) (2211111)
(3111111)
(21111111)
MATHEMATICA
nmsi[ms_]:=Select[Union[ms], Count[ms, #]<Max@@Length/@Split[ms]&];
DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[nmsi[#]]==k&]], {n, 0, 15}, {k, 0, Sqrt[n]}], 0, {2}]
CROSSREFS
Row sums are A000041.
Row lengths are approximately A000196.
Column k = 0 is A047966.
For modes we have A362614, rank statistic A362611.
For co-modes we have A362615, rank statistic A362613.
Columns k > 1 sum to A363124.
Column k = 1 is A363125.
This rank statistic (number of non-modes) is A363127.
For non-co-modes we have A363130, rank statistic A363131.
A008284/A058398 count partitions by length/mean.
A275870 counts collapsible partitions.
A353836 counts partitions by number of distinct run-sums.
A359893 counts partitions by median.
Sequence in context: A110630 A343321 A238883 * A363130 A325242 A257053
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 16 2023
STATUS
approved