A363130 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-co-modes, all 0's removed.

1, 1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 9, 10, 12, 11, 18, 1, 15, 24, 3, 13, 37, 6, 25, 43, 9, 19, 64, 18, 29, 81, 25, 33, 99, 44, 42, 129, 59, 1, 39, 162, 93, 3, 62, 201, 116, 6, 55, 247, 175, 13, 81, 303, 224, 19, 84, 364, 309, 35, 103, 457, 389, 53, 105, 535, 529, 86

Offset: 0

Author

Gus Wiseman, May 18 2023

Keywords

• nonn
• tabf

Comments

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

Examples

			Triangle begins:
   1
   1
   2
   3
   4   1
   4   3
   8   3
   6   9
  10  12
  11  18   1
  15  24   3
  13  37   6
  25  43   9
  19  64  18
  29  81  25
  33  99  44
Row n = 9 counts the following partitions:
  (9)          (441)       (32211)
  (54)         (522)
  (63)         (711)
  (72)         (3222)
  (81)         (3321)
  (333)        (4221)
  (432)        (4311)
  (531)        (5211)
  (621)        (6111)
  (222111)     (22221)
  (111111111)  (33111)
               (42111)
               (51111)
               (321111)
               (411111)
               (2211111)
               (3111111)
               (21111111)
		

Crossrefs

Row sums are A000041.

Row lengths are approximately A000196.

Column k = 0 is A047966.

For modes instead of non-co-modes we have A362614, rank stat A362611.

For co-modes instead of non-co-modes we have A362615, rank stat A362613.

For non-modes instead of non-co-modes we have A363126, rank stat A363127.

Columns k > 1 sum to A363128.

Column k = 1 is A363129.

This rank statistic (number of non-co-modes) is A363131.

A008284/A058398 count partitions by length/mean.

A275870 counts collapsible partitions.

A353836 counts partitions by number of distinct run-sums.

A359893 counts partitions by median.

Cf. A002865, A053263, A098859, A237984, A327472, A353863, A362612, A363124, A363125.

Programs

Mathematica

ncomsi[ms_]:=Select[Union[ms],Count[ms,#]>Min@@Length/@Split[ms]&];
DeleteCases[Table[Length[Select[IntegerPartitions[n] , Length[ncomsi[#]]==k&]],{n,0,15},{k,0,Sqrt[n]}],0,{2}]