A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

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

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..50.

Example

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)

Mathematica

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}]

Prog

(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

Crossrefs

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.

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, comps A353860.

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.

A353861 counts distinct weak run-sums of prime indices.

A353932 lists run-sums of standard compositions.

Rulers: A103295, A103300, A169942, A325768.

Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.

Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.

Sequence in context: A274149 A026928 A238588 * A102464 A082538 A035939

Adjacent sequences: A353860 A353861 A353862 * A353864 A353865 A353866

Keyword

nonn

Author

Gus Wiseman, Jun 04 2022

Extensions

a(31) onwards from Andrew Howroyd, Jan 15 2024

Status

approved