A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

0, 1, 1, 1, 2, 3, 2, 4, 4, 5, 7, 10, 8, 13, 13, 15, 19, 25, 24, 35, 35, 43, 50, 61, 59, 79, 83, 98, 111, 137, 137, 176, 187, 219, 240, 284, 298, 360, 385, 444, 485, 568, 600, 706, 763, 867, 951, 1088, 1168, 1345, 1453, 1641, 1792, 2023, 2179, 2467, 2673, 2988

Offset: 0

Author

Gus Wiseman, Jun 06 2023

Keywords

• nonn

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

			The a(1) = 1 through a(10) = 7 partitions:
  1  11  111  211   221    21111   2221     22211     22221      33211
              1111  2111   111111  22111    221111    32211      222211
                    11111          211111   2111111   2211111    322111
                                   1111111  11111111  21111111   2221111
                                                      111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(9) = 5 through a(12) = 8 partitions:
  (22221)      (33211)       (33221)        (2222211)
  (32211)      (222211)      (222221)       (3222111)
  (2211111)    (322111)      (322211)       (3321111)
  (21111111)   (2221111)     (332111)       (32211111)
  (111111111)  (22111111)    (2222111)      (222111111)
               (211111111)   (3221111)      (2211111111)
               (1111111111)  (22211111)     (21111111111)
                             (221111111)    (111111111111)
                             (2111111111)
                             (11111111111)
		

Crossrefs

For parts instead of multiplicities we have A087897, complement A000009.

For multisets instead of partitions we have A105039, complement A363224.

The complement is counted by A363264.

For mode we have A363484, complement A363485.

A000041 counts integer partitions, A000009 covering an initial interval.

A097979 counts normal multisets with a unique mode, complement A363262.

A362607 counts partitions with multiple modes, co-modes A362609.

A362608 counts partitions with a unique mode, co-mode A362610.

A362614 counts partitions by number of modes, co-modes A362615.

Cf. A002865, A008284, A025147, A096765, A117989, A243737, A362612.

Programs

Mathematica

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]==1&]]],{n,0,30}]