A237685 -id:A237685 - cp's OEIS frontend

A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2

Offset: 0

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), with length a(11) = 3.

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 trajectory of 94685 and the a(94685) = 5 corresponding compositions:
  94685: (2,1,1,4,1,1,2,1,1,2,1)
  86357: (2,2,4,2,2,2,2,1)
  69889: (4,4,8,1)
  65793: (8,8,1)
  65537: (16,1)

Positions of first appearances are A072639.
Positions of 1's are A333489, counted by A003242 (complement A261983).
The version for partitions is A353841.
The last part of the same trajectory is A353855.
This is the rank statistic counted by A353859.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A333627 represents the run-lengths of standard compositions.
A353832 represents the run-sum transformation of a partition.
A353840-A353846 pertain to the partition run-sum trajectory.
A353847 represents the run-sum transformation of a composition.
A353853-A353859 pertain to the composition run-sum trajectory.
A353932 lists run-sums of standard compositions, represented by A353847.
Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353849, A353850, A353852, A353860.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[FixedPointList[Total/@Split[#]&,stc[n]]]-1,{n,0,100}]

A353855 Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 6, 4, 8, 9, 8, 8, 12, 13, 8, 8, 16, 17, 18, 18, 20, 17, 22, 20, 24, 25, 24, 24, 20, 17, 18, 16, 32, 33, 34, 34, 32, 37, 38, 32, 40, 41, 32, 34, 44, 45, 32, 40, 48, 49, 50, 50, 52, 49, 54, 52, 40, 41, 40, 32, 32, 37, 34, 32, 64, 65, 66, 66, 68

Offset: 0

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, corresponding to (2,1,1) -> (2,2) -> (4), has last term a(11) = 8.

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 trajectory 139 -> 138 -> 136 -> 128 ends with a(139) = 128.

The version for partitions is A353842.
This trajectory has length A353854, firsts A072639, partitions A353841.
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 a partition's run-sum trajectory.
A353847 represents a composition's run-sums, partitions A353832.
A353853-A353859 pertain to a composition's run-sum trajectory.
Cf. A237685, A304442, A333381, A333489, A333755, A353848, A353849, A353850, A353852, A353860.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[2^Accumulate[Reverse[FixedPoint[Total/@Split[#]&,stc[n]]]]/2],{n,0,100}]

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}]

A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

Gus Wiseman, May 26 2022

nonn

The run-sums transformation is described by Kimberling at A237685 and A237750.
The runs of a sequence are its maximal consecutive constant subsequences. For example, the runs of {1,1,1,2,2,3,4} are {1,1,1}, {2,2}, {3}, {4}, with sums {3,3,4,4}.
Note that the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so this sequence lists Heinz numbers of partitions whose run-sum trajectory reaches an empty set or singleton.

			The terms together with their prime indices begin:
      1: {}            25: {3,3}           64: {1,1,1,1,1,1}
      2: {1}           27: {2,2,2}         67: {19}
      3: {2}           29: {10}            71: {20}
      4: {1,1}         31: {11}            73: {21}
      5: {3}           32: {1,1,1,1,1}     79: {22}
      7: {4}           37: {12}            81: {2,2,2,2}
      8: {1,1,1}       40: {1,1,1,3}       83: {23}
      9: {2,2}         41: {13}            84: {1,1,2,4}
     11: {5}           43: {14}            89: {24}
     12: {1,1,2}       47: {15}            97: {25}
     13: {6}           49: {4,4}          101: {26}
     16: {1,1,1,1}     53: {16}           103: {27}
     17: {7}           59: {17}           107: {28}
     19: {8}           61: {18}           109: {29}
     23: {9}           63: {2,2,4}        112: {1,1,1,1,4}
The trajectory 60 -> 45 -> 35 ends in a nonprime number 35, so 60 is not in the sequence.
The trajectory 84 -> 63 -> 49 -> 19 ends in a prime number 19, so 84 is in the sequence.

This sequence is a subset of A300273, counted by A275870.
The version for compositions is A353857, counted by A353847.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A325268 counts partitions by omicron, rank statistic A304465.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
A353853-A353859 pertain to composition run-sum trajectory.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A005811, A073093, A130091, A181819, A182857, A304660, A325239, A325277, A353839, A353862, A353867.

ope[n_]:=Times@@Prime/@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k];
Select[Range[100],#==1||PrimeQ[NestWhile[ope,#,!SquareFreeQ[#]&]]&]

A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 5, 2, 1, 0, 0, 2, 12, 2, 0, 0, 0, 8, 10, 12, 2, 0, 0, 0, 2, 32, 23, 6, 1, 0, 0, 0, 20, 26, 51, 28, 3, 0, 0, 0, 0, 5, 66, 109, 52, 22, 2, 0, 0, 0, 0, 8, 108, 144, 188, 53, 10, 1, 0, 0, 0, 0, 2, 134, 358, 282, 196, 48, 4, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 01 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 transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,1,1,3,1,1,2,1,1,2,1) -> (2,2,3,2,2,2,2,1) -> (4,3,8,1) is counted under T(15,4).

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   5   2   1   0
   0   2  12   2   0   0
   0   8  10  12   2   0   0
   0   2  32  23   6   1   0   0
   0  20  26  51  28   3   0   0   0
   0   5  66 109  52  22   2   0   0   0
   0   8 108 144 188  53  10   1   0   0   0
   0   2 134 358 282 196  48   4   0   0   0   0
For example, row n = 6 counts the following compositions:
  .  (6)       (15)     (123)    (1212)  .  .
     (33)      (24)     (132)    (2121)
     (222)     (42)     (141)
     (1113)    (51)     (213)
     (2112)    (114)    (231)
     (3111)    (411)    (312)
     (11211)   (1122)   (321)
     (111111)  (2211)   (1131)
               (11112)  (1221)
               (21111)  (1311)
                        (11121)
                        (12111)

Row sums are A011782.
Row-lengths without zeros appear to be A131737.
The version for partitions is A353843.
The length of the trajectory is A353854, firsts A072639, partitions A353841.
The last part of the same trajectory is A353855.
Column k = 1 is A353858.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333489 ranks anti-runs, counted by A003242 (complement A261983).
A333627 ranks the run-lengths of standard compositions.
A353840-A353846 pertain to partition run-sum trajectory.
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, A325277, A333755, A353848, A353850, A353852.

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

A237978 Number of partitions of n having depth 3; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 3, 1, 10, 6, 6, 13, 15, 20, 30, 30, 64, 61, 85, 99, 153, 170, 184, 271, 359, 416, 539, 631, 799, 971, 1199, 1433, 1803, 2153, 2692, 3173, 3882, 4544, 5610, 6663, 8090, 9565, 11467, 13590, 16305, 19184, 22933, 26973

Offset: 1

Clark Kimberling, Feb 19 2014

Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P),..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is introduced here as the depth of P. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.

			a(14) = 3 counts these partitions: 64211, 632111, 433211.
Successive applications of f to the first of these partitions are indicated by 64211 -> 6422 -> 644 -> 86.

Cf. A237685, A237750, A366063, A000009, A000041.

z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
Table[Count[c[n], 1], {n, 1, z}] (* A237685 *)
Table[Count[c[n], 2], {n, 1, z}] (* A237750 *)
Table[Count[c[n], 3], {n, 1, z}] (* this sequence *)
(* Peter J. C. Moses, Feb 19 2014 *)

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}]

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&]

A366063 Irregular triangle read by rows: T(n,k) is the number of partitions of n that have depth k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 0, 4, 6, 1, 5, 9, 1, 6, 11, 4, 1, 8, 20, 2, 0, 10, 25, 7, 0, 12, 37, 6, 1, 15, 47, 13, 2, 18, 67, 15, 1, 22, 85, 25, 3, 27, 122, 26, 1, 32, 142, 46, 10, 1, 38, 200, 53, 6, 0, 46, 259, 74, 6, 0, 54, 330, 92, 13, 1, 64, 412, 136

Offset: 1

Clark Kimberling, Sep 28 2023

Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P), ..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is as the depth of P, as defined in A237685. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.

			First 20 rows:
   1
   1      1
   2      1
   2      2     1
   3      4     0
   4      6     1
   5      9     1
   6     11     4    1
   8     20     2    0
  10     25     7    0
  12     37     6    1
  15     47    13    2
  18     67    15    1
  22     85    25    3
  27    122    26    1
  32    142    46   10    1
  38    200    53    6    0
  46    259    74    6    0
  54    330    92   13    1
  64    412   136   15    0

Cf. A000009, A000041, A237685 (column 1), A237750 (column 2), A237978 (column 3), A225485 (frequency depth array).

z = 36; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
t = Table[Count[c[n], k], {n, 1, z}, {k, 0, Floor[Log[2, n]]}]
TableForm[t] (* this sequence as an array *)
Flatten[t]   (* this sequence *)

A238003 Number of partitions of n not having depth 1; see Comments.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 11, 10, 17, 19, 30, 34, 50, 54, 89, 97, 126, 160, 215, 254, 339, 409, 549, 649, 838, 997, 1286, 1562, 1934, 2375, 2966, 3552, 4418, 5339, 6505, 7869, 9591, 11499, 13946, 16781, 20163, 24167, 28932, 34434, 41285, 49116, 58508, 69361

Offset: 1

Clark Kimberling, Feb 19 2014

The depth of a partition is defined at A237685 as follows. Suppose that P is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P), ..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is the depth of P. Conjecture: lim_{n->infinity} a(n)/A000041(n) = 1.

			The 11 partitions of 6 are partitioned by depth as follows:
  depth 0: 6, 51, 42, 321;
  depth 1: 411, 33, 222, 2211, 21111, 11111;
  depth 2: 3111.
Thus, a(6) = 5.

Cf. A237685, A000041.

z = 40; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
Table[Count[c[n], 1], {n, 1, z}] (* A237685 *)
Table[PartitionsP[n] - Count[c[n], 1], {n, 1, z}]  (* A238003 *)
(* Peter J. C. Moses, Feb 19 2014 *)

a(n) = A000041(n) - A237685(n) for n >= 1.

cp's OEIS Frontend

A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

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A353855 Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

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

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A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

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A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.

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A237978 Number of partitions of n having depth 3; see Comments.

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

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A366063 Irregular triangle read by rows: T(n,k) is the number of partitions of n that have depth k.

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