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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.
5

%I #5 May 18 2023 08:33:26

%S 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,

%T 18,29,81,25,33,99,44,42,129,59,1,39,162,93,3,62,201,116,6,55,247,175,

%U 13,81,303,224,19,84,364,309,35,103,457,389,53,105,535,529,86

%N 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.

%C 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}.

%e Triangle begins:

%e 1

%e 1

%e 2

%e 3

%e 4 1

%e 4 3

%e 8 3

%e 6 9

%e 10 12

%e 11 18 1

%e 15 24 3

%e 13 37 6

%e 25 43 9

%e 19 64 18

%e 29 81 25

%e 33 99 44

%e Row n = 9 counts the following partitions:

%e (9) (441) (32211)

%e (54) (522)

%e (63) (711)

%e (72) (3222)

%e (81) (3321)

%e (333) (4221)

%e (432) (4311)

%e (531) (5211)

%e (621) (6111)

%e (222111) (22221)

%e (111111111) (33111)

%e (42111)

%e (51111)

%e (321111)

%e (411111)

%e (2211111)

%e (3111111)

%e (21111111)

%t ncomsi[ms_]:=Select[Union[ms],Count[ms,#]>Min@@Length/@Split[ms]&];

%t DeleteCases[Table[Length[Select[IntegerPartitions[n] , Length[ncomsi[#]]==k&]],{n,0,15},{k,0,Sqrt[n]}],0,{2}]

%Y Row sums are A000041.

%Y Row lengths are approximately A000196.

%Y Column k = 0 is A047966.

%Y For modes instead of non-co-modes we have A362614, rank stat A362611.

%Y For co-modes instead of non-co-modes we have A362615, rank stat A362613.

%Y For non-modes instead of non-co-modes we have A363126, rank stat A363127.

%Y Columns k > 1 sum to A363128.

%Y Column k = 1 is A363129.

%Y This rank statistic (number of non-co-modes) is A363131.

%Y A008284/A058398 count partitions by length/mean.

%Y A275870 counts collapsible partitions.

%Y A353836 counts partitions by number of distinct run-sums.

%Y A359893 counts partitions by median.

%Y Cf. A002865, A053263, A098859, A237984, A327472, A353863, A362612, A363124, A363125.

%K nonn,tabf

%O 0,3

%A _Gus Wiseman_, May 18 2023