A359397 -id:A359397 - cp's OEIS frontend

A358194 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partial sums summing to k, where k ranges from n to n(n+1)/2.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Offset: 0

Gus Wiseman, Dec 31 2022

The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).

			Triangle begins:
  1
  1
  1 1
  1 0 1 1
  1 0 1 1 0 1 1
  1 0 0 1 1 0 1 1 0 1 1
  1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1
  1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
  1 0 0 0 1 1 1 1 0 1 1 1 2 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
For example, the T(15,59) = 5 partitions are: (8,2,2,2,1), (7,3,3,1,1), (6,5,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1).

Row sums are A000041.
The version for compositions is A053632.
Row lengths are A152947.
The version for reversed partitions is A264034.
A048793 = partial sums of reversed standard compositions, sum A029931.
A358134 = partial sums of standard compositions, sum A359042.
A358136 = partial sums of prime indices, sum A318283.
A359361 = partial sums of reversed prime indices, sum A304818.
Cf. A000009, A325362, A358137, A359397.

Table[Length[Select[IntegerPartitions[n],Total[Accumulate[#]]==k&]],{n,0,8},{k,n,n*(n+1)/2}]

A320387 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 3, 5, 7, 4, 7, 8, 6, 8, 11, 7, 9, 13, 9, 11, 16, 12, 15, 18, 13, 17, 20, 17, 21, 24, 19, 24, 30, 22, 28, 34, 26, 34, 38, 30, 37, 43, 37, 42, 48, 41, 50, 58, 48, 55, 64, 53, 64, 71, 59, 73, 81, 69, 79, 89, 79, 90, 101, 87, 100, 111

Offset: 0

Seiichi Manyama, Oct 12 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
Generating function of the "second integrals" of partitions: given a partition (p_1, ..., p_s) written in weakly decreasing order, write the sequence B = (b_1, b_2, ..., b_s) = (p_1, p_1 + p_2, ..., p_1 + ... + p_s).  The sequence gives the coefficients of the generating function summing q^(b_1 + ... + b_s) over all partitions of all nonnegative integers. - William J. Keith, Apr 23 2022
From Gus Wiseman, Jan 17 2023: (Start)
Equivalently, a(n) is the number of multisets (weakly increasing sequences of positive integers) with weighted sum n. For example, the Heinz numbers of the a(0) = 1 through a(15) = 7 multisets are:
  1  2  3  4  7  6   8   10  15  12  16  18  20  26  24  28
           5     11  9   17  19  14  21  22  27  41  30  32
                     13          23  29  31  33  55  39  34
                                 25      35  37      43  45
                                             49      77  47
                                                         65
                                                         121
These multisets are counted by A264034. The reverse version is A007294. The zero-based version is A359678.
(End)

			There are a(29) = 15 such partitions of 29:
  01: [29]
  02: [10, 19]
  03: [11, 18]
  04: [12, 17]
  05: [13, 16]
  06: [14, 15]
  07: [5, 10, 14]
  08: [6, 10, 13]
  09: [6, 11, 12]
  10: [7, 10, 12]
  11: [8, 10, 11]
  12: [3, 6, 9, 11]
  13: [5, 7, 8, 9]
  14: [2, 4, 6, 8, 9]
  15: [3, 5, 6, 7, 8]
There are a(30) = 18 such partitions of 30:
  01: [30]
  02: [10, 20]
  03: [11, 19]
  04: [12, 18]
  05: [13, 17]
  06: [14, 16]
  07: [5, 10, 15]
  08: [6, 10, 14]
  09: [6, 11, 13]
  10: [7, 10, 13]
  11: [7, 11, 12]
  12: [8, 10, 12]
  13: [3, 6, 9, 12]
  14: [9, 10, 11]
  15: [4, 7, 9, 10]
  16: [2, 4, 6, 8, 10]
  17: [6, 7, 8, 9]
  18: [4, 5, 6, 7, 8]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..300 from Seiichi Manyama)

Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320388.
Number of appearances of n > 0 in A304818, reverse A318283.
A053632 counts compositions by weighted sum.
A358194 counts partitions by weighted sum, reverse A264034.
Weighted sum of prime indices: A359497, A359676, A359682, A359754, A359755.
Cf. A029931, A359361, A359397, A359674, A359677, A359678.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
Table[Length[Select[Range[2^n],ots[prix[#]]==n&]],{n,10}] (* Gus Wiseman, Jan 17 2023 *)

seq(n)={Vec(sum(k=1, (sqrtint(8*n+1)+1)\2, my(t=binomial(k,2)); x^t/prod(j=1, k-1, 1 - x^(t-binomial(j,2)) + O(x^(n-t+1)))))} \\ Andrew Howroyd, Jan 22 2023

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary << 0
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0
  }
  cnt
end
def A320387(n)
  (0..n).map{|i| f(i)}
end
p A320387(50)

G.f.: Sum_{k>=1} x^binomial(k,2)/Product_{j=1..k-1} (1 - x^(binomial(k,2)-binomial(j,2))). - Andrew Howroyd, Jan 22 2023

A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Dec 30 2022

The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The partition with Heinz number n is the reversed n-th row of A112798.

			Triangle begins:
   2: 1
   3: 2
   4: 1 2
   5: 3
   6: 2 3
   7: 4
   8: 1 2 3
   9: 2 4
  10: 3 4
  11: 5
  12: 2 3 4
  13: 6
  14: 4 5
  15: 3 5
  16: 1 2 3 4
For example, the integer partition with Heinz number 90 is (3,2,2,1), so row n = 90 is (3,5,7,8).

Row-lengths are A001222.
The version for standard compositions is A048793, non-reversed A358134.
Last element in each row is A056239.
First element in each row is A061395
Rows are the partial sums of rows of A296150.
Row-sums are A304818.
A reverse version is A358136, row sums A318283, Heinz numbers A358137.
The sorted Heinz numbers of rows are A359397.
A000041 counts partitions, strict A000009.
A112798 lists prime indices, product A003963.
A355536 lists differences of prime indices.
Cf. A000720, A001221, A055396, A261079, A325362.

T:= n-> ListTools[PartialSums](sort([seq(numtheory
       [pi](i[1])$i[2], i=ifactors(n)[2])], `>`))[]:
seq(T(n), n=2..50);  # Alois P. Heinz, Jan 01 2023

Table[Accumulate[Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,2,30}]

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A358194 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partial sums summing to k, where k ranges from n to n(n+1)/2.

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A320387 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.

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A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

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