A363485 Number of integer partitions of n covering an initial interval of positive integers with more than one mode.

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 3, 1, 2, 6, 5, 3, 8, 4, 8, 11, 13, 9, 17, 17, 19, 25, 24, 23, 44, 35, 39, 54, 55, 63, 83, 79, 86, 104, 119, 125, 157, 164, 178, 220, 237, 251, 297, 324, 357, 413, 439, 486, 562, 607, 673, 765, 828, 901, 1040, 1117, 1220

Offset: 0

Author

Gus Wiseman, Jun 06 2023

Keywords

• nonn

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

			The a(n) partitions for n = {3, 6, 12, 15, 16, 18}:
  (21)  (321)   (332211)    (54321)       (443221)    (4433211)
        (2211)  (3222111)   (433221)      (3332221)   (5432211)
                (22221111)  (443211)      (4332211)   (43332111)
                            (33222111)    (33322111)  (333222111)
                            (322221111)   (43222111)  (333321111)
                            (2222211111)              (3322221111)
                                                      (32222211111)
                                                      (222222111111)
		

Crossrefs

For parts instead of multiplicities we have A025147, complement A096765.

For co-mode we have A363264, complement A363263.

The complement is counted by A363484.

A000041 counts integer partitions, A000009 covering an initial interval.

A071178 counts maxima in prime factorization, modes A362611.

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, A105039, A117989, A243737, A362612.

Programs

Mathematica

Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]>1&]]],{n,0,30}]