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Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Count[#, Max@@#]>1&]], {n, 15}]

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.

Cf. A008284, A105039, A117989, `A238478, `A238479, `A360071, A362607, A362608, A362612, A362614.

For If all parts appearing appear more than once we have A240085, for partitions A007690.

For If the greatest part appears exactly twice instead of more than once we have A243737.

For least instead of greatest we have A363224, see triangle A238342.

A008284/A058398/A327482 count partitions by mean.

A071178 counts maxima A067029 gives last exponent in prime factorization, modes A362611first A071178.

A275870 counts collapsible partitions.

Cf. A053263 ptns_medn_eq_max, A105039, A117989, A237984 ptns_mean_is_part, A238478 ptns_unq_medn, A238479 ptns_not_unq_medn, A240850 strptns_w_mean, A240851 strptns_wo_mean, A327472 ptns_wo_mean, A360071 tet_ptns_len_numdstnct, A362612 ptns_max_is_only_mode.

Cf. A008284, A105039, A117989, `A238478, `A238479, `A360071, A362612.

Cf. A053263 ptns_medn_eq_max, A105039, A117989, A237984 ptns_mean_is_part, A238478 ptns_unq_medn, A238479 ptns_not_unq_medn, A240850 strptns_w_mean, A240851 strptns_wo_mean, A327472 ptns_wo_mean, A360071 tet_ptns_len_numdstnct, A362612 ptns_max_is_only_mode.

allocated for Gus WisemanNumber of integer compositions of n in which the greatest part appears more than once.

0, 1, 1, 2, 4, 9, 18, 37, 73, 145, 287, 570, 1134, 2264, 4526, 9061, 18152, 36374, 72884, 146011, 292416, 585422, 1171632, 2344136, 4688821, 9376832, 18749169, 37485358, 74939850, 149813328, 299492966, 598729533, 1196987066, 2393137399, 4784846896, 9567357951

1,4

Also the number of multisets of length n covering an initial interval of positive integers with more than one mode.

The a(2) = 1 through a(6) = 9 compositions:

(11) (111) (22) (122) (33)

(1111) (212) (222)

(221) (1122)

(11111) (1212)

(1221)

(2112)

(2121)

(2211)

(111111)

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Count[#, Max@@#]>1&]], {n, 15}]

For partitions instead of compositions we have A002865.

The complement is counted by A097979 shifted left.

Row sums of columns k > 1 of A238341.

For all parts appearing more than once we have A240085, for partitions A007690.

For exactly twice instead of more than once we have A243737.

For least we have A363224, see triangle A238342.

A000041 counts integer partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A032020 counts strict compositions.

A071178 counts maxima in prime factorization, modes A362611.

A261982 counts compositions with some part appearing more than once.

A275870 counts collapsible partitions.

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.

Cf. A053263 ptns_medn_eq_max, A237984 ptns_mean_is_part, A238478 ptns_unq_medn, A238479 ptns_not_unq_medn, A240850 strptns_w_mean, A240851 strptns_wo_mean, A327472 ptns_wo_mean, A360071 tet_ptns_len_numdstnct, A362612 ptns_max_is_only_mode.

allocated

nonn

Gus Wiseman, Jun 04 2023

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editing

allocated for Gus Wiseman

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