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A363130 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.

%I A363130 #5 May 18 2023 08:33:26
%S A363130 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 A363130 18,29,81,25,33,99,44,42,129,59,1,39,162,93,3,62,201,116,6,55,247,175,
%U A363130 13,81,303,224,19,84,364,309,35,103,457,389,53,105,535,529,86
%N A363130 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 A363130 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 A363130 Triangle begins:
%e A363130    1
%e A363130    1
%e A363130    2
%e A363130    3
%e A363130    4   1
%e A363130    4   3
%e A363130    8   3
%e A363130    6   9
%e A363130   10  12
%e A363130   11  18   1
%e A363130   15  24   3
%e A363130   13  37   6
%e A363130   25  43   9
%e A363130   19  64  18
%e A363130   29  81  25
%e A363130   33  99  44
%e A363130 Row n = 9 counts the following partitions:
%e A363130   (9)          (441)       (32211)
%e A363130   (54)         (522)
%e A363130   (63)         (711)
%e A363130   (72)         (3222)
%e A363130   (81)         (3321)
%e A363130   (333)        (4221)
%e A363130   (432)        (4311)
%e A363130   (531)        (5211)
%e A363130   (621)        (6111)
%e A363130   (222111)     (22221)
%e A363130   (111111111)  (33111)
%e A363130                (42111)
%e A363130                (51111)
%e A363130                (321111)
%e A363130                (411111)
%e A363130                (2211111)
%e A363130                (3111111)
%e A363130                (21111111)
%t A363130 ncomsi[ms_]:=Select[Union[ms],Count[ms,#]>Min@@Length/@Split[ms]&];
%t A363130 DeleteCases[Table[Length[Select[IntegerPartitions[n] , Length[ncomsi[#]]==k&]],{n,0,15},{k,0,Sqrt[n]}],0,{2}]
%Y A363130 Row sums are A000041.
%Y A363130 Row lengths are approximately A000196.
%Y A363130 Column k = 0 is A047966.
%Y A363130 For modes instead of non-co-modes we have A362614, rank stat A362611.
%Y A363130 For co-modes instead of non-co-modes we have A362615, rank stat A362613.
%Y A363130 For non-modes instead of non-co-modes we have A363126, rank stat A363127.
%Y A363130 Columns k > 1 sum to A363128.
%Y A363130 Column k = 1 is A363129.
%Y A363130 This rank statistic (number of non-co-modes) is A363131.
%Y A363130 A008284/A058398 count partitions by length/mean.
%Y A363130 A275870 counts collapsible partitions.
%Y A363130 A353836 counts partitions by number of distinct run-sums.
%Y A363130 A359893 counts partitions by median.
%Y A363130 Cf. A002865, A053263, A098859, A237984, A327472, A353863, A362612, A363124, A363125.
%K A363130 nonn,tabf
%O A363130 0,3
%A A363130 _Gus Wiseman_, May 18 2023