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

0,5

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}.

Links

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

Formula

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

Example

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)

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]}]

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.

Sequence in context: A080844 A321428 A362615 * A336167 A076626 A182886

Adjacent sequences: A362611 A362612 A362613 * A362615 A362616 A362617

Keyword

nonn,look,tabf

Author

Gus Wiseman, May 04 2023

Status

approved