cp's OEIS Frontend

A363126 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.

%I A363126 #6 May 17 2023 23:25:55
%S A363126 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 A363126 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 A363126 72,158,87,6,55,87,214,121,13,81,95,250,177,24
%N A363126 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 A363126 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 A363126 Triangle begins:
%e A363126    1
%e A363126    1
%e A363126    2
%e A363126    3
%e A363126    4   1
%e A363126    4   3
%e A363126    8   3
%e A363126    6   8   1
%e A363126   10   9   3
%e A363126   11  13   6
%e A363126   15  18   9
%e A363126   13  24  18   1
%e A363126   25  24  25   3
%e A363126   19  36  40   6
%e A363126   29  41  52  13
%e A363126   33  45  79  19
%e A363126   42  57  95  36   1
%e A363126   39  68 133  54   3
%e A363126 Row n = 9 counts the following partitions:
%e A363126   (9)          (441)       (3321)
%e A363126   (54)         (522)       (4221)
%e A363126   (63)         (711)       (4311)
%e A363126   (72)         (3222)      (5211)
%e A363126   (81)         (6111)      (42111)
%e A363126   (333)        (22221)     (321111)
%e A363126   (432)        (32211)
%e A363126   (531)        (33111)
%e A363126   (621)        (51111)
%e A363126   (222111)     (411111)
%e A363126   (111111111)  (2211111)
%e A363126                (3111111)
%e A363126                (21111111)
%t A363126 nmsi[ms_]:=Select[Union[ms],Count[ms,#]<Max@@Length/@Split[ms]&];
%t A363126 DeleteCases[Table[Length[Select[IntegerPartitions[n],Length[nmsi[#]]==k&]],{n,0,15},{k,0,Sqrt[n]}],0,{2}]
%Y A363126 Row sums are A000041.
%Y A363126 Row lengths are approximately A000196.
%Y A363126 Column k = 0 is A047966.
%Y A363126 For modes we have A362614, rank statistic A362611.
%Y A363126 For co-modes we have A362615, rank statistic A362613.
%Y A363126 Columns k > 1 sum to A363124.
%Y A363126 Column k = 1 is A363125.
%Y A363126 This rank statistic (number of non-modes) is A363127.
%Y A363126 For non-co-modes we have A363130, rank statistic A363131.
%Y A363126 A008284/A058398 count partitions by length/mean.
%Y A363126 A275870 counts collapsible partitions.
%Y A363126 A353836 counts partitions by number of distinct run-sums.
%Y A363126 A359893 counts partitions by median.
%Y A363126 Cf. A002865, A053263, A098859, A237984, A238478, A327472, A353863, A353864, A362612.
%K A363126 nonn,tabf
%O A363126 0,3
%A A363126 _Gus Wiseman_, May 16 2023