OFFSET
0,3
COMMENTS
We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.
EXAMPLE
Triangle begins:
1
1
2
3
4 1
4 3
8 3
6 9
10 12
11 18 1
15 24 3
13 37 6
25 43 9
19 64 18
29 81 25
33 99 44
Row n = 9 counts the following partitions:
(9) (441) (32211)
(54) (522)
(63) (711)
(72) (3222)
(81) (3321)
(333) (4221)
(432) (4311)
(531) (5211)
(621) (6111)
(222111) (22221)
(111111111) (33111)
(42111)
(51111)
(321111)
(411111)
(2211111)
(3111111)
(21111111)
MATHEMATICA
ncomsi[ms_]:=Select[Union[ms], Count[ms, #]>Min@@Length/@Split[ms]&];
DeleteCases[Table[Length[Select[IntegerPartitions[n] , Length[ncomsi[#]]==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.
Columns k > 1 sum to A363128.
Column k = 1 is A363129.
This rank statistic (number of non-co-modes) is A363131.
A275870 counts collapsible partitions.
A353836 counts partitions by number of distinct run-sums.
A359893 counts partitions by median.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 18 2023
STATUS
approved