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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

%I #6 May 17 2023 23:25:55

%S 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,

%T 3,19,36,40,6,29,41,52,13,33,45,79,19,42,57,95,36,1,39,68,133,54,3,62,

%U 72,158,87,6,55,87,214,121,13,81,95,250,177,24

%N 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.

%C 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}.

%e Triangle begins:

%e 1

%e 1

%e 2

%e 3

%e 4 1

%e 4 3

%e 8 3

%e 6 8 1

%e 10 9 3

%e 11 13 6

%e 15 18 9

%e 13 24 18 1

%e 25 24 25 3

%e 19 36 40 6

%e 29 41 52 13

%e 33 45 79 19

%e 42 57 95 36 1

%e 39 68 133 54 3

%e Row n = 9 counts the following partitions:

%e (9) (441) (3321)

%e (54) (522) (4221)

%e (63) (711) (4311)

%e (72) (3222) (5211)

%e (81) (6111) (42111)

%e (333) (22221) (321111)

%e (432) (32211)

%e (531) (33111)

%e (621) (51111)

%e (222111) (411111)

%e (111111111) (2211111)

%e (3111111)

%e (21111111)

%t nmsi[ms_]:=Select[Union[ms],Count[ms,#]<Max@@Length/@Split[ms]&];

%t DeleteCases[Table[Length[Select[IntegerPartitions[n],Length[nmsi[#]]==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 we have A362614, rank statistic A362611.

%Y For co-modes we have A362615, rank statistic A362613.

%Y Columns k > 1 sum to A363124.

%Y Column k = 1 is A363125.

%Y This rank statistic (number of non-modes) is A363127.

%Y For non-co-modes we have A363130, rank statistic 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, A238478, A327472, A353863, A353864, A362612.

%K nonn,tabf

%O 0,3

%A _Gus Wiseman_, May 16 2023