A353845 -id:A353845 - cp's OEIS frontend

A353846 Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.

1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 1, 0, 0, 0, 0, 5, 9, 1, 0, 0, 0, 0, 0, 6, 11, 4, 1, 0, 0, 0, 0, 0, 8, 20, 2, 0, 0, 0, 0, 0, 0, 0, 10, 25, 7, 0, 0, 0, 0, 0, 0, 0, 0, 12, 37, 6, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, May 26 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 run-sums (or condensations) until a strict partition is reached. For example, the trajectory of (2,1,1) is (2,1,1) -> (2,2) -> (4).

Also the number of integer partitions of n with Kimberling's depth statistic (see A237685, A237750) equal to k-1.

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   1   0
   0   2   2   1   0
   0   3   4   0   0   0
   0   4   6   1   0   0   0
   0   5   9   1   0   0   0   0
   0   6  11   4   1   0   0   0   0
   0   8  20   2   0   0   0   0   0   0
   0  10  25   7   0   0   0   0   0   0   0
   0  12  37   6   1   0   0   0   0   0   0   0
   0  15  47  13   2   0   0   0   0   0   0   0   0
   0  18  67  15   1   0   0   0   0   0   0   0   0   0
   0  22  85  25   3   0   0   0   0   0   0   0   0   0   0
   0  27 122  26   1   0   0   0   0   0   0   0   0   0   0   0
For example, row n = 8 counts the following partitions (empty columns indicated by dots):
.  (8)    (44)        (422)     (4211)  .  .  .  .
   (53)   (332)       (32111)
   (62)   (611)       (41111)
   (71)   (2222)      (221111)
   (431)  (3221)
   (521)  (3311)
          (5111)
          (22211)
          (311111)
          (2111111)
          (11111111)

Row-sums are A000041.
Column k = 1 is A000009.
Column k = 2 is A237685.
Column k = 3 is A237750.
The version for run-lengths instead of run-sums is A225485 or A325280.
This statistic (trajectory length) is ranked by A353841 and A326371.
The version for compositions is A353859, see also A353847-A353858.
A005811 counts runs in binary expansion.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A353832 represents the operation of taking run-sums of a partition
A353836 counts partitions by number of distinct run-sums.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
A353845 counts partitions whose run-sum trajectory ends in a singleton.
Cf. A008284, A047966, A181819, A325239, A325277, A353834, A353865.

rsn[y_]:=If[y=={},{},NestWhileList[Reverse[Sort[Total/@ Split[Sort[#]]]]&,y,!UnsameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],Length[rsn[#]]==k&]],{n,0,15},{k,0,n}]

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

Original entry on oeis.org

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}]

A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3

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.
Starting with n, this is one plus the number of times one must apply A353832 to reach a squarefree number.
Also Kimberling's depth statistic (defined in A237685 and A237750) plus one.

			The trajectory for a(1080) = 4 is the following, with prime indices shown on the right:
  1080: {1,1,1,2,2,2,3}
   325: {3,3,6}
   169: {6,6}
    37: {12}
The trajectory for a(87780) = 5 is the following, 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 trajectory for a(39960) = 5 is the following, with prime indices shown on the right:
  39960: {1,1,1,2,2,2,3,12}
  12025: {3,3,6,12}
   6253: {6,6,12}
   1369: {12,12}
     89: {24}

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of 1's are A005117.
The version for run-lengths instead of sums is A182850 or A323014.
Positions of first appearances are A353743.
These are the row-lengths of A353840.
Other sequences pertaining to this trajectory are A353842-A353845.
Counting partitions by this statistic gives A353846.
The version for compositions is A353854, run-lengths of A353853.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A318928 gives runs-resistance of binary expansion.
A353832 represents the operation of taking run-sums of a partition.
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.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A008966, A071625, A073093, A181819, A182857, A325239, A325277, A325278, A353834, A353847, A353865, A353867.

Table[If[n==1,0,Length[NestWhileList[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&]]],{n,100}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum)));
A353841(n) = if(1==n,0,for(i=1,oo,if(issquarefree(n), return(i), n = A353832(n)))); \\ Antti Karttunen, Jan 20 2025

a(1) = 0, and for n > 1, if A008966(n) = 1 [n is in A005117], a(n) = 1, otherwise a(n) = 1+a(A353832(n)). [See comments] - Antti Karttunen, Jan 20 2025

More terms from Antti Karttunen, Jan 20 2025

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}]

A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 2, 5, 5, 5, 1, 2, 12, 1, 8, 11, 3, 3, 19, 8, 5, 27, 9, 1, 2, 34, 19, 1, 15, 26, 34, 2, 2, 49, 45, 5, 5, 68, 48, 14, 4, 58, 98, 15, 1, 18, 76, 105, 31, 1, 2, 88, 159, 46, 2, 13, 98, 191, 79, 4, 2, 114, 261, 105, 8, 14, 148, 282, 164, 19

Offset: 0

Gus Wiseman, Jun 04 2022

The partition run-sum trajectory is obtained by repeatedly taking the run-sums until a strict partition is reached. For example, the trajectory of y = (3,2,1,1,1) is (3,2,1,1,1) -> (3,3,2) -> (6,2), so y is counted under T(8,2).

			Triangle begins:
   1
   1
   2
   2  1
   4  1
   2  5
   5  5  1
   2 12  1
   8 11  3
   3 19  8
   5 27  9  1
   2 34 19  1
  15 26 34  2
   2 49 45  5
   5 68 48 14
   4 58 98 15  1
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (4211)      (611)
  (41111)     (3311)
  (221111)    (5111)
  (11111111)  (22211)
              (32111)
              (311111)
              (2111111)

Row sums are A000041.
Row-lengths are A003056.
The last part of the same trajectory is A353842.
Column k = 1 is A353845, compositions A353858.
The length of the trajectory is A353846.
The version for compositions is A353856.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with constant run-sums, ranked by A353833/A353834.
A325268 counts partitions by omicron, rank statistic A304465.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
A353932 lists run-sums of standard compositions.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.
Cf. A047966, A237685, A325277, A353841, A353853-A353859.

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

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A353846 Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.

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

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A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.

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

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A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

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