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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.
+10
39
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
COMMENTS
The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).
EXAMPLE
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).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Total[Accumulate[#]]==k&]], {n, 0, 8}, {k, n, n*(n+1)/2}]
CROSSREFS
The version for compositions is A053632. The version for reversed partitions is A264034. A048793 = partial sums of reversed standard compositions, sum A029931.
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.
+10
35
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
COMMENTS
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 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)
FORMULA
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
EXAMPLE
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]
MATHEMATICA
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 *)
PROG
(Ruby)
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
(0..n).map{|i| f(i)}
end
(PARI) 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
CROSSREFS
A053632 counts compositions by weighted sum.
Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.
+10
28
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
COMMENTS
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.
EXAMPLE
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).
MAPLE
T:= n-> ListTools[PartialSums](sort([seq(numtheory
[pi](i[1])$i[2], i=ifactors(n)[2])], `>`))[]:
MATHEMATICA
Table[Accumulate[Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 2, 30}]
CROSSREFS
The version for standard compositions is A048793, non-reversed A358134. Last element in each row is A056239. First element in each row is A061395Rows are the partial sums of rows of A296150. The sorted Heinz numbers of rows are A359397. A355536 lists differences of prime indices.
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