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

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, 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, 6, 1, 0, 95, 81, 44, 10, 1, 0, 123, 105, 53, 14, 2

Offset: 0

Author

Gus Wiseman, May 04 2023

Keywords

• nonn
• tabf

Comments

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}.

Examples

			Triangle begins:
   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  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   6   1
   0  95  81  44  10   1
Row n = 8 counts the following partitions:
  (8)         (53)     (431)
  (44)        (62)     (521)
  (332)       (71)
  (422)       (3221)
  (611)       (3311)
  (2222)      (4211)
  (5111)      (32111)
  (22211)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)
		

Links

• Alois P. Heinz, Rows n = 0..800, flattened

Crossrefs

Row sums are A000041.

Row lengths are A002024.

Removing columns 0 and 1 and taking sums gives A362609, ranks A362606.

Column k = 1 is A362610, ranks A359178.

This statistic (co-mode count) is ranked by A362613.

For mode instead of co-mode we have A362614, ranked by A362611.

A008284 counts partitions by length.

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

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

A275870 counts collapsible partitions.

Cf. A003056, A098859, A325347, A359893, A362607, A362608, A362612, A372632.

Programs

Mathematica

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[comsi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Formula

Sum_{k=0..A003056(n)} k * T(n,k) = A372632(n). - Alois P. Heinz, May 07 2024