A353844 -id:A353844 - cp's OEIS frontend

A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

1, 2, 3, 4, 3, 5, 6, 7, 8, 5, 9, 7, 10, 11, 12, 9, 7, 13, 14, 15, 16, 7, 17, 18, 14, 19, 20, 15, 21, 22, 23, 24, 15, 25, 13, 26, 27, 13, 28, 21, 29, 30, 31, 32, 11, 33, 34, 35, 36, 21, 37, 38, 39, 40, 25, 13, 41, 42, 43, 44, 33, 45, 35, 46, 47, 48, 21, 49, 19

Offset: 1

Gus Wiseman, May 25 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 given in row 12 corresponds to the partitions (2,1,1) -> (2,2) -> (4).
This is the iteration of the transformation f described by Kimberling at A237685.

			Triangle begins:
   1
   2
   3
   4  3
   5
   6
   7
   8  5
   9  7
  10
  11
  12  9  7
Row 87780 is the following trajectory (left column), with prime indices shown on the right:
  87780: {1,1,2,3,4,5,8}
  65835: {2,2,3,4,5,8}
  51205: {3,4,4,5,8}
  19855: {3,5,8,8}
   2915: {3,5,16}

The version for run-lengths instead of sums is A325239 or A325277.
This is the iteration of A353832, with composition version A353847.
Row-lengths are A353841, counted by A353846.
Final terms are A353842.
Counting rows by final omega gives A353843.
Rows ending in a prime number are A353844, counted by A353845.
These sequences for compositions are A353853-A353859.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A182850 or A323014 gives frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353862 gives greatest run-sum of prime indices, least A353931.
Cf. A071625, A073093, A181819, A323023, A353743, A353834, A353864, A353865, A353866, A353867.

Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&,n,Not@*SquareFreeQ],{n,30}]

A353860 Number of collapsible integer compositions of n.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 12, 2, 26, 9, 36, 2, 206, 2, 132, 40, 677, 2, 1746, 2, 3398, 136, 2052, 2, 44388, 33, 8196, 730, 79166, 2, 263234, 2, 458330, 2056, 131076, 160, 8804349, 2, 524292, 8200, 13662156, 2, 36036674, 2, 48844526, 90282, 8388612, 2, 1971667502, 129

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

If a collapse is a joining of some number of adjacent equal parts of an integer composition, we call a composition collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (1,1,1,3,2,1,1,2) -> (3,3,2,1,1,2) -> (3,3,2,2,2) -> (6,2,2,2) -> (6,6) -> (12), which shows that (1,1,1,3,2,1,1,2) is a collapsible composition of 12.

			The a(0) = 0 through a(6) = 12 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)
          (11)  (111)  (22)    (11111)  (33)
                       (112)            (222)
                       (211)            (1113)
                       (1111)           (1122)
                                        (2112)
                                        (2211)
                                        (3111)
                                        (11112)
                                        (11211)
                                        (21111)
                                        (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for partitions is A275870, ranked by A300273.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A011782 counts compositions.
A353847 represents the run-sums of a composition, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A318928, A333755, A353844, A353848, A353849, A353850, A353852.

repcams[q_List]:=repcams[q]=Union[{q},If[UnsameQ@@q,{},Union@@repcams/@ Union[Insert[Drop[q,#],Plus@@Take[q,#],First[#]]&/@ Select[Tuples[Range[Length[q]],2],And[Less@@#,SameQ@@Take[q,#]]&]]]];
Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],MemberQ[repcams[#],{n}]&]],{n,0,15}]

a(n) = if(n==0, 0, 1 - sumdiv(n, d, if(d>1, moebius(d)*a(n/d)^d ))) \\ Andrew Howroyd, Feb 04 2023

Sum_{d|n} mu(d)*a(n/d)^d = 1 for n > 0. - Andrew Howroyd, Feb 04 2023

Terms a(16) and beyond from Andrew Howroyd, Feb 04 2023

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

Original entry on oeis.org

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

Gus Wiseman, Jun 04 2022

nonn

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.

			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)

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.

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}]

\\ 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

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

A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 8, 2, 20, 5, 8, 2, 78, 2, 8, 8, 223, 2, 179, 2, 142, 8, 8, 2, 4808

Offset: 0

Gus Wiseman, Jun 17 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sums (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).

			The a(0) = 0 through a(8) = 20 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
          (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                       (112)            (222)                (224)
                       (211)            (1113)               (422)
                       (1111)           (2112)               (1124)
                                        (3111)               (2114)
                                        (11211)              (2222)
                                        (111111)             (4112)
                                                             (4211)
                                                             (11114)
                                                             (21122)
                                                             (22112)
                                                             (41111)
                                                             (111122)
                                                             (112112)
                                                             (211211)
                                                             (221111)
                                                             (1111211)
                                                             (1121111)
                                                             (11111111)

The version for partitions is A353845, ranked by A353844.
The trajectory itself is A353853, last part A353855.
The lengths of trajectories of standard compositions are A353854.
This is column k = 1 of A353856, for partitions A353843.
These compositions are ranked by A353857.
A011782 counts compositions.
A066099 lists compositions in standard order.
A238279 and A333755 count compositions by number of runs.
A275870 counts collapsible partitions, ranked by A300273.
A333489 ranks anti-runs, counted by A003242 (complement A261983).
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353851 counts compositions with equal run-sums, ranked by A353848.
A353859 counts compositions by length of run-sum trajectory.
A353860 counts collapsible compositions.
A353932 lists run-sums of standard compositions.
Cf. A072639, A237685, A304442, A304465, A318928, A353833, A353841, A353849, A353850, A353852.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&,#]]==1&]],{n,0,15}]

A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 174, 175, 184, 186, 187, 232, 238, 239, 248, 250, 251, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047

Offset: 1

Gus Wiseman, Jun 01 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   7:      111  (1,1,1)
   8:     1000  (4)
  10:     1010  (2,2)
  11:     1011  (2,1,1)
  14:     1110  (1,1,2)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  31:    11111  (1,1,1,1,1)
  32:   100000  (6)
  36:   100100  (3,3)
  39:   100111  (3,1,1,1)
  42:   101010  (2,2,2)
  46:   101110  (2,1,1,2)
  59:   111011  (1,1,2,1,1)
  60:   111100  (1,1,1,3)
  63:   111111  (1,1,1,1,1,1)

The version for partitions is A353844.
The trajectory length is A353854, firsts A072639, for partitions A353841.
The last part of the trajectory is A353855, for partitions A353842.
These compositions are counted by A353858.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333627 ranks the run-lengths of standard compositions.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents composition run-sum transformation, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353848, A353849, A353850, A353852.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],Length[FixedPoint[Total/@Split[#]&,stc[#]]]==1&]

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A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

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A353860 Number of collapsible integer compositions of n.

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

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A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

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A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

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