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%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