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

%I A353863 #10 Jan 15 2024 20:29:11
%S A353863 1,1,1,2,2,3,4,6,7,10,11,16,20,24,30,43,47,62,79,94,113,143,170,211,
%T A353863 256,307,372,449,531,648,779,926,1100,1323,1562,1864,2190,2595,3053,
%U A353863 3611,4242,4977,5834,6825,7973,9344,10844,12641,14699,17072,19822
%N A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.
%C A353863 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}.
%C A353863 This is a kind of completeness property, cf. A126796.
%e A353863 The a(1) = 1 through a(8) = 7 partitions:
%e A353863   (1)  (11)  (21)   (211)   (311)    (321)     (3211)     (3221)
%e A353863              (111)  (1111)  (2111)   (3111)    (4111)     (32111)
%e A353863                             (11111)  (21111)   (22111)    (41111)
%e A353863                                      (111111)  (31111)    (221111)
%e A353863                                                (211111)   (311111)
%e A353863                                                (1111111)  (2111111)
%e A353863                                                           (11111111)
%t A353863 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A353863 msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
%t A353863 wkrs[y_]:=Union[Total/@Select[msubs[y],SameQ@@#&]];
%t A353863 Table[Length[Select[IntegerPartitions[n],normQ[Rest[wkrs[#]]]&]],{n,0,15}]
%o A353863 (PARI) \\ isok(p) tests the partition.
%o A353863 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}
%o A353863 a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ _Andrew Howroyd_, Jan 15 2024
%Y A353863 For parts instead of weak run-sums we have A000009.
%Y A353863 For multiplicities instead of weak run-sums we have A317081.
%Y A353863 If weak run-sums are distinct we have A353865, the completion of A353864.
%Y A353863 A003242 counts anti-run compositions, ranked by A333489, complement A261983.
%Y A353863 A005811 counts runs in binary expansion.
%Y A353863 A165413 counts distinct run-lengths in binary expansion, sums A353929.
%Y A353863 A300273 ranks collapsible partitions, counted by A275870, comps A353860.
%Y A353863 A353832 represents taking run-sums of a partition, compositions A353847.
%Y A353863 A353833 ranks partitions with all equal run-sums, counted by A304442.
%Y A353863 A353835 counts distinct run-sums of prime indices.
%Y A353863 A353837 counts partitions with distinct run-sums, ranked by A353838.
%Y A353863 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353863 A353861 counts distinct weak run-sums of prime indices.
%Y A353863 A353932 lists run-sums of standard compositions.
%Y A353863 Rulers: A103295, A103300, A169942, A325768.
%Y A353863 Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.
%Y A353863 Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.
%K A353863 nonn
%O A353863 0,4
%A A353863 _Gus Wiseman_, Jun 04 2022
%E A353863 a(31) onwards from _Andrew Howroyd_, Jan 15 2024