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A353863
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Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.
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21
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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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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}.
This is a kind of completeness property, cf. A126796.
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LINKS
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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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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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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 weak run-sums are distinct we have A353865, the completion of A353864.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A353832 represents taking run-sums of a partition, compositions A353847.
A353835 counts distinct run-sums of prime indices.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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