Revision History for A367411
(Underlined text is an addition;
strikethrough text is a deletion.)
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#5 by Michael De Vlieger at Sat Nov 18 18:07:58 EST 2023
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#4 by Gus Wiseman at Sat Nov 18 17:07:28 EST 2023
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#3 by Gus Wiseman at Sat Nov 18 17:05:13 EST 2023
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| CROSSREFS
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The complement forFor parts instead of sums we ishave A001227.A238007:
The complement for all subset-sums is A188431, non-strict A126796 (ranks A325781).
For parts instead of sums we have A238007
For all subset-sums we have A365831, non-strict A365924 (ranks A365830).
- complement A001227
- non-strict complement A034296, ranks A073491
- non-strict A239955, ranks A073492
The complement is counted by A367410, non-strict version is A367402A367403.
The non-strict versioncomplement is A367403, complement A367402.
The complement is counted by A367410.
The non-binary version is A365831:
- non-strict complement A126796, ranks A325781
- complement A188431
- non-strict A365924, ranks A365830
A034296 counts partitions covering an interval, ranks A073491.
A239955 counts partitions not covering an interval, ranks A073492.
A261036 counts complete partitions by maximum.
A276024 counts positive subset-sums of partitions, strict A284640.
`Cf. A000041, ~, A002033, ~A080259, A261036, A264401, `, A276024, A284640, A304792, `, A364272, `A365658, ~A365918, `A365921.
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#2 by Gus Wiseman at Fri Nov 17 23:48:46 EST 2023
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| NAME
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allocatedNumber of strict integer partitions of n whose semi-sums do not cover an interval forof Guspositive Wisemanintegers.
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| DATA
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0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 4, 5, 8, 10, 14, 16, 23, 27, 35, 42, 52, 61, 75, 89, 106, 126, 149, 173, 204, 237, 274, 319, 369, 424, 490, 560, 642, 734, 838, 952, 1085, 1231, 1394, 1579, 1784, 2011, 2269, 2554, 2872, 3225, 3619, 4054, 4540, 5077, 5671, 6332
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| OFFSET
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0,9
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| COMMENTS
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We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
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| EXAMPLE
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The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is counted under a(7).
The a(7) = 1 through a(13) = 10 partitions:
(4,2,1) (4,3,1) (5,3,1) (5,3,2) (5,4,2) (6,4,2) (6,4,3)
(5,2,1) (6,2,1) (5,4,1) (6,3,2) (6,5,1) (6,5,2)
(6,3,1) (6,4,1) (7,3,2) (7,4,2)
(7,2,1) (7,3,1) (7,4,1) (7,5,1)
(8,2,1) (8,3,1) (8,3,2)
(9,2,1) (8,4,1)
(5,4,2,1) (9,3,1)
(6,3,2,1) (10,2,1)
(6,4,2,1)
(7,3,2,1)
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| MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#, {2}]; If[d=={}, {}, Range[Min@@d, Max@@d]]!=Union[d])&]], {n, 0, 30}]
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| CROSSREFS
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The complement for parts instead of sums is A001227.
The complement for all subset-sums is A188431, non-strict A126796 (ranks A325781).
For parts instead of sums we have A238007
For all subset-sums we have A365831, non-strict A365924 (ranks A365830).
The complement is counted by A367410, non-strict A367402.
The non-strict version is A367403, complement A367402.
A000009 counts partitions covering an initial interval, ranks A055932.
A034296 counts partitions covering an interval, ranks A073491.
A046663 counts partitions w/o submultiset summing to k, strict A365663.
A239955 counts partitions not covering an interval, ranks A073492.
A261036 counts complete partitions by maximum.
A276024 counts positive subset-sums of partitions, strict A284640.
A365543 counts partitions w/ submultiset summing to k, strict A365661.
`Cf. A000041, ~A002033, ~A080259, A264401, `A304792, `A364272, `A365658, ~A365918, `A365921.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Nov 17 2023
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Fri Nov 17 06:38:33 EST 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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