Revision History for A353863
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing all changes.
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#10 by Michael De Vlieger at Mon Jan 15 20:29:11 EST 2024
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#9 by Andrew Howroyd at Mon Jan 15 20:11:51 EST 2024
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#8 by Andrew Howroyd at Mon Jan 15 19:34:00 EST 2024
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| DATA
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1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779, 926, 1100, 1323, 1562, 1864, 2190, 2595, 3053, 3611, 4242, 4977, 5834, 6825, 7973, 9344, 10844, 12641, 14699, 17072, 19822
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| PROG
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(PARI) \\ isok(p) tests the partition.
isok(p)={my(b=0, s=0, t=0); for(i=1, #p, if(p[i]<>t, t=p[i]; s=0); s += t; b = bitor(b, 1<<(s-1))); bitand(b, b+1)==0}
a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ Andrew Howroyd, Jan 15 2024
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| KEYWORD
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nonn,more
nonn
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| EXTENSIONS
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a(31) onwards from Andrew Howroyd, Jan 15 2024
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| STATUS
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approved
editing
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#7 by Michael De Vlieger at Mon Jun 06 08:05:34 EDT 2022
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#6 by Gus Wiseman at Mon Jun 06 00:04:28 EDT 2022
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#5 by Gus Wiseman at Mon Jun 06 00:00:38 EDT 2022
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| COMMENTS
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This is a kind of completeness property, cf. A126796.
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| CROSSREFS
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Rulers: A103295, A103300, A169942, A325768.
Cf. A047967, A073093, A181819, A237685, `A353834, A353844, `A353848, `A353849, `A353855, `A353858, A353867, A353930.
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#4 by Gus Wiseman at Sun Jun 05 18:41:31 EDT 2022
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| CROSSREFS
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If the weak run-sums are distinct we have A353865, the complete casecompletion of A353864.
A300273 ranks collapsible partitions, counted by A275870, compositionscomps A353860.
`A353932 lists run-sums of standard compositions.
Complete: A002033 (ranked by , A325780), , A126796, A276024, A325781 (strict , A188431), , A353866.
Cf. A047967, A073093, `A116608, A181819, A237685, `A325277, `A333755, `A353834, A353844, `A353848, `A353849, `A353855, `A353858, A353867, A353930.
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#3 by Gus Wiseman at Sun Jun 05 18:35:57 EDT 2022
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| NAME
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Run-sum-complete partitions. Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.
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| COMMENTS
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This is a kind of completeness property, cf.
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| CROSSREFS
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A003242 counts anti-run compositions , ranked by A333489, complement A261983.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
`A353932 lists run-sums of standard compositions.
Rulers: A103295, A103300, A169942, A325768
Complete: A002033 (ranked by A325780), A126796, A276024, A325781 (strict A188431), A353866.
Cf. A047967, A071625, A073093, , `A116608, A175413, A181819, A237685, A238279, , `A325277, A333381, , `A333755, `A353834, A353839, A353844, , `A353848, , `A353849, , `A353855, , `A353858, A353866, A353867, A353930.
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#2 by Gus Wiseman at Sat Jun 04 02:32:55 EDT 2022
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| NAME
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allocatedRun-sum-complete partitions. Number of integer partitions of n whose weak run-sums cover an initial interval forof Gusnonnegative Wisemanintegers.
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| DATA
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1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779
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| OFFSET
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0,4
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| COMMENTS
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A weak run-sum of a sequence is the sum of any consecutive constant subsequence. For example, the weak run-sums of (3,2,2,1) are {1,2,3,4}.
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| EXAMPLE
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The a(1) = 1 through a(8) = 7 partitions:
(1) (11) (21) (211) (311) (321) (3211) (3221)
(111) (1111) (2111) (3111) (4111) (32111)
(11111) (21111) (22111) (41111)
(111111) (31111) (221111)
(211111) (311111)
(1111111) (2111111)
(11111111)
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| MATHEMATICA
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normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t, i], {t, Split[s]}, {i, 0, Length[t]}]];
wkrs[y_]:=Union[Total/@Select[msubs[y], SameQ@@#&]];
Table[Length[Select[IntegerPartitions[n], normQ[Rest[wkrs[#]]]&]], {n, 0, 15}]
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| CROSSREFS
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For parts instead of weak run-sums we have A000009.
For multiplicities instead of weak run-sums we have A317081.
If the weak run-sums are distinct we have A353865, the complete case of A353864.
A003242 counts anti-run compositions ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870, compositions A353860.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
Cf. A047967, A071625, A073093, A116608, A175413, A181819, A237685, A238279, A325277, A333381, A333755, `A353834, A353839, A353844, A353848, A353849, A353855, A353858, A353866, A353867, A353930.
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| KEYWORD
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allocated
nonn,more
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| AUTHOR
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Gus Wiseman, Jun 04 2022
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Sun May 08 15:31:24 EDT 2022
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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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