A275870 -id:A275870 - cp's OEIS frontend

A289079 Number of orderless same-trees of weight n with all leaves equal to 1.

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 13, 1, 3, 3, 22, 1, 16, 1, 15, 3, 3, 1, 151, 2, 3, 6, 17, 1, 41, 1, 334, 3, 3, 3, 637, 1, 3, 3, 275, 1, 56, 1, 21, 19, 3, 1, 15591, 2, 27, 3, 23, 1, 902, 3, 516, 3, 3, 1, 7858, 1, 3, 21, 69109, 3, 98, 1, 27, 3, 67, 1, 811756, 1

Offset: 1

Gus Wiseman, Jun 23 2017

nonn

a(n) is also the number of orderless same-trees of weight n with all leaves greater than 1.

			The a(12)=13 orderless same-trees with all leaves greater than 1 are: ((33)(33)), ((33)(222)), ((33)6), ((222)(222)), ((222)6), (66), ((22)(22)(22)), ((22)(22)4), ((22)44), (444), (3333), (222222), 12.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A196545, A273873, A275870, A281145, A281146, A289078.

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1, add(
      binomial(a(n/d)+d-1, d), d=divisors(n) minus {1}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jul 05 2017

a[n_]:=If[n===1,1,Sum[Binomial[a[n/d]+d-1,d],{d,Rest[Divisors[n]]}]];
Array[a,100]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = sumdiv(n, d, binomial(v[n/d]+d-1, d))); v} \\ Andrew Howroyd, Aug 20 2018

from sympy import divisors, binomial
l=[0, 1]
for n in range(2, 101): l+=[sum([binomial(l[n//d] + d - 1, d) for d in divisors(n)[1:]]), ]
l[1:] # Indranil Ghosh, Jul 06 2017

a(1) = 1, a(n>1) = Sum_{d|n, d>1} binomial(a(n/d)+d-1, d).

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 20, 30, 32, 56, 64, 90, 128, 140, 176, 210, 256, 416, 512, 616, 990, 1024, 1088, 1540, 2048, 2288, 2310, 2432, 2970, 4096, 4950, 5888, 7072, 7700, 8008, 8192, 11550, 12870, 14848, 16384, 20020, 20672, 30030, 31744, 32768, 38896, 50490, 55936

Offset: 1

Gus Wiseman, Jun 07 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.
Related concepts:
- A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).
- In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.
- A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).
- A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   32: {1,1,1,1,1}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  140: {1,1,3,4}
  176: {1,1,1,1,5}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}

Knapsack partitions are counted by A108917, ranked by A299702.
Complete partitions are counted by A126796, ranked by A325781.
These partitions are counted by A353865.
This is a special case of A353866, counted by A353864, complement A354583.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]

A357875 Numbers whose run-sums of prime indices are weakly increasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Oct 18 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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 prime indices of 24 are (1,1,1,2), with run-sums (3,2), which are not weakly increasing, so 24 is not in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

These partitions are counted by A304405.
These are the indices  of rows in A354584 that are weakly increasing.
The complement is A357876.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A047966, A118914, A181819, A239312, A275870, A300273, A304442, A325249, A353743-A354912.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],LessEqual@@Total/@Split[primeMS[#]]&]

A304405 Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 52, 61, 80, 97, 127, 147, 189, 220, 277, 325, 402, 469, 578, 665, 804, 933, 1121, 1282, 1537, 1754, 2081, 2374, 2793, 3179, 3739, 4232, 4923, 5587, 6477, 7305, 8445, 9519, 10949, 12323, 14110, 15825, 18099, 20229, 23005

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with weakly decreasing run-sums, complement A357878. - Gus Wiseman, Oct 22 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 2+1                  | 1, 2
  | 1+1+1                | 3
4 | 4                    | 4
  | 3+1                  | 1, 3
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 4+1                  | 1, 4
  | 3+2                  | 2, 3
  | 3+1+1                | 2, 3
  | 2+2+1                | 1, 4
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 5+1                  | 1, 5
  | 4+2                  | 2, 4
  | 4+1+1                | 2, 4
  | 3+3                  | 6
  | 3+2+1                | 1, 2, 3
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 2+2+1+1              | 2, 4
  | 1+1+1+1+1+1          | 6

The strict opposite version is A304430, ranked by A357864.
The strict version is A304428, ranked by A357862.
The opposite version is A304406, ranked by A357861.
Number of rows in A354584 summing to n that are strictly increasing.
These partitions are ranked by A357875.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100883, A239312, A275870, A353832, A353864.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 22 2022 *)

A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 5, 2, 3, 4, 3, 2, 4, 3, 3, 4, 4, 3, 4, 3, 4, 5, 5, 4, 6, 4, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 9, 6, 6, 7, 6, 8, 9, 6, 6, 8, 9, 7, 9, 9, 7, 10, 9, 8, 13, 7, 10, 11, 8, 9, 10, 11, 12, 9, 11, 9, 15, 12, 12, 19, 13, 16, 16

Offset: 0

Gus Wiseman, Jun 04 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). A weak run-sum is the sum of any consecutive constant subsequence.

Do all positive integers appear only finitely many times in this sequence?

			The a(n) compositions for n = 1, 3, 9, 15, 18:
  (1)  (21)   (4311)       (54321)            (543321)
       (111)  (51111)      (532221)           (654111)
              (111111111)  (651111)           (7611111)
                           (81111111)         (111111111111111111)
                           (111111111111111)
For example, the weak runs of y = {7,5,4,4,3,3,3,1,1} are {}, {1}, {1,1}, {3}, {4}, {5}, {3,3}, {7}, {4,4}, {3,3,3}, with sums 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are all distinct and cover an initial interval, so y is counted under a(31).

Perfect partitions are counted by A002033, ranked by A325780.
Knapsack partitions are counted by A108917, ranked by A299702.
This is the complete case of A353864, ranked by A353866.
These partitions are ranked by A353867.
A000041 counts partitions, strict A000009.
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.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353850 counts compositions with all distinct run-sums, ranked by A353852.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A008284, A018818, A225485, A325239, A325862, A353834, A353839.

norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Table[Length[Select[IntegerPartitions[n],norqQ[Total/@Select[msubs[#],SameQ@@#&]]&]],{n,0,15}]

a(n) = my(c=0, s, v); if(n, forpart(p=n, if(p[1]==1, v=List([s=1]); for(i=2, #p, if(p[i]==p[i-1], listput(v, s+=p[i]), listput(v, s=p[i]))); s=#v; listsort(v, 1); if(s==#v&&s==v[s], c++))); c, 1); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 11, 14, 20, 26, 33, 41, 50, 64, 81, 97, 120, 150, 176, 210, 255, 303, 362, 426, 503, 595, 703, 816, 953, 1113, 1283, 1482, 1721, 1988, 2299, 2650, 3031, 3464, 3965, 4492, 5115, 5820, 6592, 7467, 8484, 9568, 10822, 12185, 13724, 15445, 17381, 19475, 21855

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with strictly decreasing run-sums. - Gus Wiseman, Oct 21 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 2+1                  | 1, 2
  | 1+1+1                | 3
4 | 4                    | 4
  | 3+1                  | 1, 3
  | 2+2                  | 4
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 4+1                  | 1, 4
  | 3+2                  | 2, 3
  | 3+1+1                | 2, 3
  | 2+2+1                | 1, 4
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 5+1                  | 1, 5
  | 4+2                  | 2, 4
  | 4+1+1                | 2, 4
  | 3+3                  | 6
  | 3+2+1                | 1, 2, 3
  | 2+2+2                | 6
  | 2+2+1+1              | 2, 4
  | 1+1+1+1+1+1          | 6

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 71 terms from Seiichi Manyama)

The weak version is A304405, ranked by A357875.
The weak opposite version is A304406, ranked by A357861.
The opposite version is A304430, ranked by A357864.
Number of rows in A354584 summing to n that are strictly increasing.
These partitions are ranked by A357862, complement A357863.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100471, A239312, A275870, A353832, A353864.

Table[Length[Select[IntegerPartitions[n],Greater@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 21 2022 *)

a(n) <= A304405(n).

A304430 Number of partitions of n in which the sequence of the sum of the same summands is decreasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 10, 10, 13, 15, 18, 19, 22, 26, 33, 33, 38, 41, 50, 53, 60, 68, 77, 84, 94, 100, 116, 122, 136, 148, 172, 182, 206, 219, 246, 258, 281, 301, 341, 365, 397, 429, 466, 489, 528, 572, 623, 660, 728, 773, 849, 895, 968, 1019, 1120, 1188, 1288

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with strictly increasing run-sums. - Gus Wiseman, Oct 22 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 2+1+1+1              | 3, 2
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 2+2+2                | 6
  | 2+1+1+1+1            | 4, 2
  | 1+1+1+1+1+1          | 6

The weak opposite version is A304405, ranked by A357875.
The weak version is A304406, ranked by A357861.
The opposite version is A304428, ranked by A357862.
Number of rows in A354584 summing to n that are strictly decreasing.
These partitions are ranked by A357864.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100471, A100881, A239312, A275870, A353832, A353833, A353864.

Table[Length[Select[IntegerPartitions[n],Less@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 22 2022 *)

a(n) <= A304406(n).

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 19, 21, 30, 31, 42, 50, 62, 69, 91, 99, 126, 144, 175, 198, 246, 275, 331, 379, 452, 509, 612, 686, 811, 922, 1076, 1219, 1428, 1604, 1863, 2108, 2434, 2739, 3162, 3551, 4075, 4593, 5240, 5885, 6721, 7527, 8556, 9597, 10870

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

A000009(n) <= a(n) <= A000041(n).
Equivalently, the number of partitions of n such that (maximal multiplicity of parts) = (multiplicity of the maximal part), as in the Mathematica program. - Clark Kimberling, Apr 04 2014
Also the number of integer partitions of n whose greatest part is a mode, meaning it appears at least as many times as each of the others. The name "Number of partitions of n such that smaller parts do not occur more frequently than greater parts" seems to describe A100882 = "Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing," which first differs from this at n = 10 due to the partition (3,3,2,1,1). - Gus Wiseman, May 07 2023

			a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

For median instead of mode we have A053263.
The complement is counted by A240302.
The case where the maximum is the only mode is A362612.
A000041 counts integer partitions, strict A000009.
A362608 counts partitions with a unique mode, complement A362607.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes.
Cf. A002865, A008284, A098859, A237984, A275870, A325347, A362610.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)
(* Clark Kimberling, Apr 04 2014 *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
     If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1,
     If[k == -1, j, If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
a[n_] := PartitionsP[n] - b[n, n, -1];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz in A240302 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Commonest[#],Max[#]]&]],{n,0,30}] (* Gus Wiseman, May 07 2023 *)

{ my(N=53, x='x+O('x^N));
my(gf=1+sum(i=1,N,sum(j=1,floor(N/i),x^(i*j)*prod(k=1,i-1,(1-x^(k*(j+1)))/(1-x^k)))));
Vec(gf) } \\ John Tyler Rascoe, Mar 09 2024

a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k) +p(n,max(i,j),1,k+1) else (if j0 then 0 else 1).

a(n) + A240302(n) = A000041(n). - Clark Kimberling, Apr 04 2014.

G.f.: 1 + Sum_{i, j>0} x^(i*j) * Product_{k=1..i-1} ((1 - x^(k*(j+1)))/(1 - x^k)). - John Tyler Rascoe, Mar 09 2024

cp's OEIS Frontend

A289079 Number of orderless same-trees of weight n with all leaves equal to 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A357875 Numbers whose run-sums of prime indices are weakly increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A304405 Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A304430 Number of partitions of n in which the sequence of the sum of the same summands is decreasing.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Views

Author

Keywords

Comments