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

Author

Gus Wiseman, Jun 04 2022

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

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}]

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

Extensions

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