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

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

Offset: 0

Author

Gus Wiseman, May 04 2023

Keywords

• nonn
• look
• tabf

Comments

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}.

Examples

			Triangle begins:
   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   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   1
   0 175  29  17   9   1
Row n = 8 counts the following partitions:
  (8)         (53)    (431)
  (44)        (62)    (521)
  (332)       (71)
  (422)       (3311)
  (611)
  (2222)
  (3221)
  (4211)
  (5111)
  (22211)
  (32111)
  (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 A362607, ranks A362605.

Column k = 1 is A362608, ranks A356862.

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

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

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

Programs

Mathematica

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

Formula

Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - Alois P. Heinz, May 05 2024