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A363130
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Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-co-modes, all 0's removed.
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5
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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, 42, 129, 59, 1, 39, 162, 93, 3, 62, 201, 116, 6, 55, 247, 175, 13, 81, 303, 224, 19, 84, 364, 309, 35, 103, 457, 389, 53, 105, 535, 529, 86
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OFFSET
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0,3
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COMMENTS
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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}.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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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}]
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CROSSREFS
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Row lengths are approximately A000196.
For co-modes instead of non-co-modes we have A362615, rank stat A362613.
For non-modes instead of non-co-modes we have A363126, rank stat A363127.
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.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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