%I #20 May 05 2024 16:42:18

%S 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 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 1,0,175,29,17,9,1,0,233,33,18,11,2,0,295,47,25

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

%C 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 Alois P. Heinz, <a href="/A362614/b362614.txt">Rows n = 0..800, flattened</a>

%F Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - _Alois P. Heinz_, May 05 2024

%e Triangle begins:

%e 1

%e 0 1

%e 0 2

%e 0 2 1

%e 0 4 1

%e 0 5 2

%e 0 7 3 1

%e 0 11 3 1

%e 0 16 4 2

%e 0 21 6 3

%e 0 29 8 4 1

%e 0 43 7 5 1

%e 0 54 13 8 2

%e 0 78 12 8 3

%e 0 102 17 11 5

%e 0 131 26 12 6 1

%e 0 175 29 17 9 1

%e Row n = 8 counts the following partitions:

%e (8) (53) (431)

%e (44) (62) (521)

%e (332) (71)

%e (422) (3311)

%e (611)

%e (2222)

%e (3221)

%e (4211)

%e (5111)

%e (22211)

%e (32111)

%e (41111)

%e (221111)

%e (311111)

%e (2111111)

%e (11111111)

%t msi[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];

%t Table[Length[Select[IntegerPartitions[n],Length[msi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

%Y Row sums are A000041.

%Y Row lengths are A002024.

%Y Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.

%Y Column k = 1 is A362608, ranks A356862.

%Y This statistic (mode-count) is ranked by A362611.

%Y For co-modes we have A362615, ranked by A362613.

%Y A008284 counts partitions by length.

%Y A096144 counts partitions by number of minima, A026794 by maxima.

%Y A238342 counts compositions by number of minima, A238341 by maxima.

%Y A275870 counts collapsible partitions.

%Y Cf. A002865, A003056, A098859, A240219, A359893, A360071, A362609, A362610, A362612, A372542.

%K nonn,look,tabf

%O 0,5

%A _Gus Wiseman_, May 04 2023