A363263 - OEIS

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

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OFFSET

0,5

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}.

LINKS

Table of n, a(n) for n=0..57.

EXAMPLE

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)

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}]

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.

Sequence in context: A182762 A173997 A029143 * A153846 A284383 A072406

Adjacent sequences: A363260 A363261 A363262 * A363264 A363265 A363266

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 06 2023

STATUS

approved