Revision History for A362609
(Underlined text is an addition;
strikethrough text is a deletion.)
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#7 by Michael De Vlieger at Tue May 02 16:08:00 EDT 2023
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#6 by Gus Wiseman at Tue May 02 14:49:18 EDT 2023
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#5 by Gus Wiseman at Tue May 02 14:48:59 EDT 2023
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| CROSSREFS
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A000041 counts integer partitions.
Cf. A002865, A008284, `, A053263, A098859, A304442, `A327472, `A353836, `A353863, A353864, A360071, A362612.
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#4 by Gus Wiseman at Mon May 01 22:31:40 EDT 2023
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| COMMENTS
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Alternatively, theseThese are partitions where no part appears fewer times than all of everythe otherothers.
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| CROSSREFS
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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.
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#3 by Gus Wiseman at Mon May 01 14:49:21 EDT 2023
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| COMMENTS
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Number ofAlternatively, these integerare partitions of n where no part appears fewer times than every other.
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| EXAMPLE
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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).
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| CROSSREFS
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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.
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#2 by Gus Wiseman at Sun Apr 30 23:04:40 EDT 2023
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| NAME
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allocatedNumber of integer partitions of n with more than one part forof Gusleast Wisemanmultiplicity.
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| DATA
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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
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| OFFSET
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0,6
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| COMMENTS
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Number of integer partitions of n where no part appears fewer times than every other.
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| EXAMPLE
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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)
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| MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]>1&]], {n, 0, 30}]
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| CROSSREFS
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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.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Apr 30 2023
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Fri Apr 28 03:41:10 EDT 2023
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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