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A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

%I A362614 #20 May 05 2024 16:42:18
%S A362614 1,0,1,0,2,0,2,1,0,4,1,0,5,2,0,7,3,1,0,11,3,1,0,16,4,2,0,21,6,3,0,29,
%T A362614 8,4,1,0,43,7,5,1,0,54,13,8,2,0,78,12,8,3,0,102,17,11,5,0,131,26,12,6,
%U A362614 1,0,175,29,17,9,1,0,233,33,18,11,2,0,295,47,25
%N A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.
%C A362614 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
%H A362614 Alois P. Heinz, <a href="/A362614/b362614.txt">Rows n = 0..800, flattened</a>
%F A362614 Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - _Alois P. Heinz_, May 05 2024
%e A362614 Triangle begins:
%e A362614    1
%e A362614    0   1
%e A362614    0   2
%e A362614    0   2   1
%e A362614    0   4   1
%e A362614    0   5   2
%e A362614    0   7   3   1
%e A362614    0  11   3   1
%e A362614    0  16   4   2
%e A362614    0  21   6   3
%e A362614    0  29   8   4   1
%e A362614    0  43   7   5   1
%e A362614    0  54  13   8   2
%e A362614    0  78  12   8   3
%e A362614    0 102  17  11   5
%e A362614    0 131  26  12   6   1
%e A362614    0 175  29  17   9   1
%e A362614 Row n = 8 counts the following partitions:
%e A362614   (8)         (53)    (431)
%e A362614   (44)        (62)    (521)
%e A362614   (332)       (71)
%e A362614   (422)       (3311)
%e A362614   (611)
%e A362614   (2222)
%e A362614   (3221)
%e A362614   (4211)
%e A362614   (5111)
%e A362614   (22211)
%e A362614   (32111)
%e A362614   (41111)
%e A362614   (221111)
%e A362614   (311111)
%e A362614   (2111111)
%e A362614   (11111111)
%t A362614 msi[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A362614 Table[Length[Select[IntegerPartitions[n],Length[msi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]
%Y A362614 Row sums are A000041.
%Y A362614 Row lengths are A002024.
%Y A362614 Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.
%Y A362614 Column k = 1 is A362608, ranks A356862.
%Y A362614 This statistic (mode-count) is ranked by A362611.
%Y A362614 For co-modes we have A362615, ranked by A362613.
%Y A362614 A008284 counts partitions by length.
%Y A362614 A096144 counts partitions by number of minima, A026794 by maxima.
%Y A362614 A238342 counts compositions by number of minima, A238341 by maxima.
%Y A362614 A275870 counts collapsible partitions.
%Y A362614 Cf. A002865, A003056, A098859, A240219, A359893, A360071, A362609, A362610, A362612, A372542.
%K A362614 nonn,look,tabf
%O A362614 0,5
%A A362614 _Gus Wiseman_, May 04 2023