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A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.

%I A362615 #15 May 07 2024 19:42:09
%S A362615 1,0,1,0,2,0,2,1,0,4,1,0,5,2,0,7,3,1,0,10,4,1,0,13,7,2,0,16,11,3,0,23,
%T A362615 14,4,1,0,30,19,6,1,0,35,29,11,2,0,50,34,14,3,0,61,46,23,5,0,73,69,27,
%U A362615 6,1,0,95,81,44,10,1,0,123,105,53,14,2
%N A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.
%C A362615 We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.
%H A362615 Alois P. Heinz, <a href="/A362615/b362615.txt">Rows n = 0..800, flattened</a>
%F A362615 Sum_{k=0..A003056(n)} k * T(n,k) = A372632(n). - _Alois P. Heinz_, May 07 2024
%e A362615 Triangle begins:
%e A362615    1
%e A362615    0   1
%e A362615    0   2
%e A362615    0   2   1
%e A362615    0   4   1
%e A362615    0   5   2
%e A362615    0   7   3   1
%e A362615    0  10   4   1
%e A362615    0  13   7   2
%e A362615    0  16  11   3
%e A362615    0  23  14   4   1
%e A362615    0  30  19   6   1
%e A362615    0  35  29  11   2
%e A362615    0  50  34  14   3
%e A362615    0  61  46  23   5
%e A362615    0  73  69  27   6   1
%e A362615    0  95  81  44  10   1
%e A362615 Row n = 8 counts the following partitions:
%e A362615   (8)         (53)     (431)
%e A362615   (44)        (62)     (521)
%e A362615   (332)       (71)
%e A362615   (422)       (3221)
%e A362615   (611)       (3311)
%e A362615   (2222)      (4211)
%e A362615   (5111)      (32111)
%e A362615   (22211)
%e A362615   (41111)
%e A362615   (221111)
%e A362615   (311111)
%e A362615   (2111111)
%e A362615   (11111111)
%t A362615 comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
%t A362615 Table[Length[Select[IntegerPartitions[n],Length[comsi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]
%Y A362615 Row sums are A000041.
%Y A362615 Row lengths are A002024.
%Y A362615 Removing columns 0 and 1 and taking sums gives A362609, ranks A362606.
%Y A362615 Column k = 1 is A362610, ranks A359178.
%Y A362615 This statistic (co-mode count) is ranked by A362613.
%Y A362615 For mode instead of co-mode we have A362614, ranked by A362611.
%Y A362615 A008284 counts partitions by length.
%Y A362615 A096144 counts partitions by number of minima, A026794 by maxima.
%Y A362615 A238342 counts compositions by number of minima, A238341 by maxima.
%Y A362615 A275870 counts collapsible partitions.
%Y A362615 Cf. A003056, A098859, A325347, A359893, A362607, A362608, A362612, A372632.
%K A362615 nonn,tabf
%O A362615 0,5
%A A362615 _Gus Wiseman_, May 04 2023