Revision History for A237821
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#14 by Michael De Vlieger at Sat May 27 10:33:02 EDT 2023
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#13 by Seiichi Manyama at Sat May 27 10:16:21 EDT 2023
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#12 by Seiichi Manyama at Sat May 27 09:55:25 EDT 2023
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| FORMULA
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G.f.: sum{Sum_{i>=1} sum{Sum_{j>=0} x^(3*i+j)/prod() /Product_{k=i, ..2*i+j, } (1-x^k))). - _Seiichi Manyama_, May 27 2023
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#11 by Seiichi Manyama at Sat May 27 09:54:11 EDT 2023
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| FORMULA
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G.f.: sum{i>=1} sum{j>=0} x^(3*i+j)/prod(k=i, 2*i+j, 1-x^k))
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| STATUS
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approved
editing
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#10 by Michael De Vlieger at Thu May 18 11:54:03 EDT 2023
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#9 by Gus Wiseman at Wed May 17 23:26:28 EDT 2023
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#8 by Gus Wiseman at Mon May 15 22:12:18 EDT 2023
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#7 by Gus Wiseman at Mon May 15 22:08:10 EDT 2023
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| EXAMPLE
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(21) (31) (41) (42) (52) (62)
(211) (221) (51) (61) (71)
(311) (321) (331) (422)
(2111) (411) (421) (431)
(2211) (511) (521)
(3111) (2221) (611)
(21111) (3211) (3221)
(4111) (3311)
(22111) (4211)
(31111) (5111)
(211111) (22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(311) (321) (61) (71)
(2111) (411) (322) (422)
(2211) (421) (431)
(3111) (511) (521)
(21111) (3211) (611)
(4111) (3221)
(22111) (3311)
(31111) (4211)
(211111) (5111)
(32111)
(41111)
(221111)
(311111)
(2111111)
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| CROSSREFS
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Cf. A237820.
The conjugate partitions have ranks A362980.
Cf. A002865 ptns_no1, A008284 tri_ptns, A171979 ptns_max_is_mode, A237984 ptns_mean_is_part, A238478 ptns_unq_medn, A238479 ptns_not_unq_medn, A327472 ptns_wo_mean, A359893 tri_ptns_by_medn_half, A362612 ptns_max_is_only_mode, A362622 prix_max_in_mid, A362980 prix_medn_neq_max.
Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.
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#6 by Gus Wiseman at Mon May 15 22:03:11 EDT 2023
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| COMMENTS
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By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023
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| EXAMPLE
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From Gus Wiseman, May 15 2023: (Start)
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
(21) (31) (41) (42) (52) (62)
(211) (221) (51) (61) (71)
(311) (321) (331) (422)
(2111) (411) (421) (431)
(2211) (511) (521)
(3111) (2221) (611)
(21111) (3211) (3221)
(4111) (3311)
(22111) (4211)
(31111) (5111)
(211111) (22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(311) (321) (61) (71)
(2111) (411) (322) (422)
(2211) (421) (431)
(3111) (511) (521)
(21111) (3211) (611)
(4111) (3221)
(22111) (3311)
(31111) (4211)
(211111) (5111)
(32111)
(41111)
(221111)
(311111)
(2111111)
(End)
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| CROSSREFS
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The complement is counted by A053263, ranks A081306.
These partitions have ranks A069900.
The case of equality is A118096.
For > instead of <= we have A237820, ranks A362982.
For <= instead of >= we have A237824, ranks A362981.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865 ptns_no1, A008284 tri_ptns, A171979 ptns_max_is_mode, A237984 ptns_mean_is_part, A238478 ptns_unq_medn, A238479 ptns_not_unq_medn, A327472 ptns_wo_mean, A359893 tri_ptns_by_medn_half, A362612 ptns_max_is_only_mode, A362622 prix_max_in_mid, A362980 prix_medn_neq_max.
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| STATUS
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approved
editing
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#5 by Michel Marcus at Tue Feb 18 13:05:20 EST 2014
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