proposed

approved

editing

proposed

A000041 counts integer partitions.

Cf. A002865, A008284, `, A053263, A098859, A304442, `A327472, `A353836, `A353863, A353864, A360071, A362612.

Alternatively, theseThese are partitions where no part appears fewer times than all of everythe otherothers.

For parts instead of multiplicities we have A117989, ranks A283050.

A000041 counts integer partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A304442A359893 counts partitions with all equalby run-sumsmedian.

`A325347 counts partitions with integer median, complement A307683.

`A353836 counts partitions by number of distinct run-sums.

`A359893/A359901/A359902 count partitions by median.

Cf. `. A002865, ~`, A008284, `A053263, A098859, `~A237984, `~A240219, `~A327472, ~`A327476, A304442, `A327472, `A353836, `A353863, A353864, A353865, A360071, `A360687, A362612.

Number ofAlternatively, these integerare partitions of n where no part appears fewer times than every other.

The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1, (namely 1 and 4, ), so is counted under a(9).

For mode instead of co-mode we have A362607 (, ranks A362605), complement A362608 (ranks A356862)..

For mode complement instead of co-mode we have A362608, ranks A356862.

`A325347 counts partitions with integer median, strict A359907, ranked by A359908, complement A307683.

`A353836 counts partitions by number of distinct run-sums.

`A359893/A359901/A359902 count partitions by median.

A362611 counts modes in prime factorization, co-modes A362613.

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

Cf. A002577 bin_ptns_of_pow2, A002865 ptns_multimax, A018818 ptns_use_divs, A051293 non0_subs_int_mean, A053263 ptns_medn_eq_max, A098859 ptns_dstnct_mults, A108917 ks, A145515 tri_ptns_of_k_to_n_into_pow_k, A237984 ptns_mean_is_part, A240219 ptns_mean_eq_medn, A240850 strptns_w_mean, A240851 strptns_wo_mean, A327472 ptns_wo_mean, A327473 h_mean_is_part, A327474 numof_dstnct_means_of_subs, A327475 subs_int_mean, A327476 h_ptns_wo_mean, A353863 ptns_run_complete, A353864 ptns_rucksack, A353865 ptns_rucksack_perfect, A360071 tet_ptns_len_numdstnct, A360686 ptns_dstnctpts_int_medn, A360687 ptns_mults_int_medn, A360952 strptns_nonint_medn, A362612 ptns_max_is_only_mode, A362613 ptns_unimode_is_max.

Cf. `A002865, ~`A053263, A098859, `~A237984, `~A240219, `~A327472, ~`A327476, A353863, A353864, A353865, A360071, `A360687, A362612.

allocatedNumber of integer partitions of n with more than one part forof Gusleast Wisemanmultiplicity.

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 19, 26, 42, 51, 74, 103, 136, 174, 246, 303, 411, 523, 674, 844, 1114, 1364, 1748, 2174, 2738, 3354, 4247, 5139, 6413, 7813, 9613, 11630, 14328, 17169, 20958, 25180, 30497, 36401, 44025, 52285, 62834, 74626, 89111, 105374, 125662

0,6

Number of integer partitions of n where no part appears fewer times than every other.

The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1, namely 1 and 4, so is counted under a(9).

The a(3) = 1 through a(9) = 14 partitions:

(21) (31) (32) (42) (43) (53) (54)

(41) (51) (52) (62) (63)

(321) (61) (71) (72)

(2211) (421) (431) (81)

(3211) (521) (432)

(3221) (531)

(3311) (621)

(4211) (3321)

(32111) (4221)

(4311)

(5211)

(42111)

(222111)

(321111)

Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]>1&]], {n, 0, 30}]

For median instead of co-mode we have A238479, complement A238478.

These partitions have ranks A362606.

For mode instead of co-mode we have A362607 (ranks A362605), complement A362608 (ranks A356862).

The complement is counted by A362610, ranks A359178.

A000041 counts integer partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A275870 counts collapsible partitions.

A304442 counts partitions with all equal run-sums.

A325347 counts partitions with integer median, strict A359907, ranked by A359908, complement A307683.

A353836 counts partitions by number of distinct run-sums.

A359893/A359901/A359902 count partitions by median.

A362611 counts modes in prime factorization.

A362614 counts partitions by number of modes.

Cf. A002577 bin_ptns_of_pow2, A002865 ptns_multimax, A018818 ptns_use_divs, A051293 non0_subs_int_mean, A053263 ptns_medn_eq_max, A098859 ptns_dstnct_mults, A108917 ks, A145515 tri_ptns_of_k_to_n_into_pow_k, A237984 ptns_mean_is_part, A240219 ptns_mean_eq_medn, A240850 strptns_w_mean, A240851 strptns_wo_mean, A327472 ptns_wo_mean, A327473 h_mean_is_part, A327474 numof_dstnct_means_of_subs, A327475 subs_int_mean, A327476 h_ptns_wo_mean, A353863 ptns_run_complete, A353864 ptns_rucksack, A353865 ptns_rucksack_perfect, A360071 tet_ptns_len_numdstnct, A360686 ptns_dstnctpts_int_medn, A360687 ptns_mults_int_medn, A360952 strptns_nonint_medn, A362612 ptns_max_is_only_mode, A362613 ptns_unimode_is_max.

allocated

nonn

Gus Wiseman, Apr 30 2023

approved

editing

allocated for Gus Wiseman

allocated

approved