A258259 -id:A258259 - cp's OEIS frontend

A111133 Number of partitions of n into at least two distinct parts.

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 26, 31, 37, 45, 53, 63, 75, 88, 103, 121, 141, 164, 191, 221, 255, 295, 339, 389, 447, 511, 584, 667, 759, 863, 981, 1112, 1259, 1425, 1609, 1815, 2047, 2303, 2589, 2909, 3263, 3657, 4096, 4581, 5119, 5717, 6377

Offset: 0

David Sharp (davidsharp(AT)rcn.com), Oct 17 2005

Old name: Number of sets of natural numbers less than n which sum to n.
From Clark Kimberling, Mar 13 2012: (Start)
(1) Number of partitions of n into at least two distinct parts.
(2) Also, number of partitions of 2n into distinct parts having maximal part n; see Example section. (End)

			a(6) = 3 because 1+5, 2+4 and 1+2+3 each sum to 6. That is, the three sets are {1,5},{2,4},{1,2,3}.
For n=6, the partitions of 2n into distinct parts having maximum 6 are 6+5+1, 6+4+2, 6+3+2+1, so that a(6)=3, as an example for Comment (2). - _Clark Kimberling_, Mar 13 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Thomas Enkosky and Branden Stone, Sequences defined by h-vectors, arXiv:1308.4945 [math.CO], 2013.
Harmandeep Kaur and Muhammad Asif Rana, Partitions with unique largest part and their generating functions, arXiv:2506.11447 [math.CO], 2025. See p. 6.

Cf. A058377.
Cf. A000009.
Cf. A026906, A258259.

a111133 = subtract 1 . a000009  -- Reinhard Zumkeller, Sep 09 2015

seq(coeff(series(mul((1+x^k),k=1..n)-1/(1-x), x,n+1),x,n),n=0..60); # Muniru A Asiru, Aug 10 2018

Needs["DiscreteMath`Combinatorica`"]
f[n_] := Block[{lmt = Floor[(Sqrt[8n + 1] - 1)/2] + 1, t}, Sum[ Length[ Select[Plus @@@ KSubsets[ Range[n - k(k - 1)/2 + 1], k], # == n &]], {k, 2, lmt}]]; Array[f, 55] (* Robert G. Wilson v, Oct 17 2005 *)
(* Next program shows the partitions (sets) *)
d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@ #] == 1 &]; Table[d[n], {n, 1, 12}]
TableForm[%]
(* Clark Kimberling, Mar 13 2012 *)
Table[PartitionsQ[n]-1, {n, 0, 55}] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *)

N=66;  x='x+O('x^N);
gf=sum(k=0,N, x^((k^2+k)/2) / prod(j=1,k, 1-x^j)) - 1/(1-x);
concat( [0,0,0], Vec(gf) ) /* Joerg Arndt, Sep 17 2012 */

a(n) = A000009(n) - 1. - Vladeta Jovovic, Oct 19 2005
G.f.: Sum_{k>=0} (x^((k^2+k)/2) / Product_{j=1..k} (1-x^j)) - 1/(1-x). - Joerg Arndt, Sep 17 2012
a(n) = A026906(floor(n-1)/2) + A258259(n). - Bob Selcoe, Oct 05 2015
G.f.: -1/(1 - x) + Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Aug 10 2018
G.f.: Sum_{n >= 1} x^(2*n+1)/Product_{k = 1..n+1} 1 - x^(2*k-1). - Peter Bala, Nov 20 2024

More terms from Vladeta Jovovic and Robert G. Wilson v, Oct 17 2005

a(0)=0 prepended by Joerg Arndt, Sep 17 2012

A368484 Number of compositions (ordered partitions) of n into parts not greater than n/2.

Original entry on oeis.org

1, 0, 1, 1, 5, 8, 24, 44, 108, 208, 464, 912, 1936, 3840, 7936, 15808, 32192, 64256, 129792, 259328, 521472, 1042432, 2091008, 4180992, 8375296, 16748544, 33525760, 67047424, 134156288, 268304384, 536739840, 1073463296, 2147205120, 4294377472, 8589344768

Offset: 0

Ilya Gutkovskiy, Dec 26 2023

Index entries for sequences related to compositions
Index entries for linear recurrences with constant coefficients, signature (2,4,-8,-4,8).

Cf. A011782, A110618, A258259.

CoefficientList[Series[(1 - 2 x - 3 x^2 + 7 x^3 + 3 x^4 - 6 x^5)/((1 - 2 x) (1 - 2 x^2)^2), {x, 0, 34}], x]
Join[{1}, LinearRecurrence[{2, 4, -8, -4, 8}, {0, 1, 1, 5, 8}, 34]]

G.f.: (1 - 2*x - 3*x^2 + 7*x^3 + 3*x^4 - 6*x^5) / ((1 - 2*x) * (1 - 2*x^2)^2).

a(n) = [x^n] 1 / (1 - Sum_{1 <= j <= n/2} x^j).

A368501 Number of compositions (ordered partitions) of n into distinct parts not greater than n/2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 6, 0, 6, 6, 36, 30, 66, 60, 120, 234, 318, 432, 666, 894, 1272, 2226, 2772, 3960, 5496, 7524, 10068, 13776, 22488, 27756, 39162, 51264, 70398, 91386, 124152, 158574, 247554, 301656, 416748, 537690, 730854, 929196, 1248798, 1576014, 2078328, 2956110

Offset: 0

Ilya Gutkovskiy, Dec 27 2023

nonn

			a(6) = 6 because we have [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2] and [3,2,1].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A008289, A032020, A072575, A110618, A258259, A368484.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n, iquo(n, 2), 0):
seq(a(n), n=0..45);  # Alois P. Heinz, Dec 28 2023

Table[Sum[Count[IntegerPartitions[n, {k}], _?(And[UnsameQ @@ #, AllTrue[#, # <= n/2 &]] &)] k!, {k, 0, n}], {n, 0, 45}]

a(n) = Sum_{k=1..floor(n/2)} A072575(n,k) for n>=1. - Alois P. Heinz, Dec 31 2023

A262885 Irregular triangle T(n,k) read by rows: T(n,k) = number of partitions of n into at least two distinct parts, where the largest part is n-k.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 1, 2, 2, 3, 3, 2, 1, 1, 2, 2, 3, 4, 3, 1, 1, 1, 2, 2, 3, 4, 4, 3, 1, 1, 1, 2, 2, 3, 4, 5, 4, 3, 1, 1, 1, 2, 2, 3, 4, 5, 5, 5, 3, 1, 1, 2, 2, 3, 4, 5, 6, 6, 5, 2, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 5, 2

Offset: 1

Bob Selcoe, Oct 04 2015

Alternate name: T(n,k) = the number of ways that at least two distinct positive integers sum to n, where the largest of these integers is n-k.
Row sums = A111133(n).
Row sums {k <= floor((n-1)/2)} = A026906(n)
Row sums {k > floor((n-1)/2)} = A258259(n)

			Triangle starts T(1,1):
n/k  1 2 3 4 5 6 7 8 9 10 11 12 13 14
1    0
2    0
3    1
4    1
5    1 1
6    1 1 1
7    1 1 2
8    1 1 2 1
9    1 1 2 2 1
10   1 1 2 2 2 1
11   1 1 2 2 3 2
12   1 1 2 2 3 3 2
13   1 1 2 2 3 4 3 1
14   1 1 2 2 3 4 4 3 1
15   1 1 2 2 3 4 5 4 3 1
16   1 1 2 2 3 4 5 5 5 3
17   1 1 2 2 3 4 5 6 6 5  2
18   1 1 2 2 3 4 5 6 7 7  5  2
19   1 1 2 2 3 4 5 6 8 8  7  5  1
20   1 1 2 2 3 4 5 6 8 9  9  8  4  1
T(15,8) = 4: the four partitions of 15 into at least two distinct parts with largest part 15-8 = 7 are  {7,6,2}; {7,5,3}; {7,5,2,1} and {7,4,3,1}.
T(14,k) for k=1..F, with F = floor(13/2) = 6: T(14,1) = 0+1 = 1; T(14,2) = 0+1 = 1; T(14,3) = 1+1 = 2; T(14,4) = 1+1 = 2; T(14,5) = 2+1 = 3; T(14,6) = 3+1 = 4.
T(14,k) for k>6: T(14,7) = T(7,1)+T(7,2)+T(7,3) = 1+1+2 = 4; T(14,8) = T(8,3)+T(8,4) = 2+1 = 3; T(14,9) = T(9,5) = 1.

Cf. A000009, A026906, A111133, A258259.

Given T(1,1) = T(2,1) = 0, to find row n>=3: Let k" be the maximum value of k in row g

T(n,k) = S(g)+1 g=k when g<=F (equivalent to A000009(g));

T(n,k) = Sum_{j=2*(g-F)-1..k"} T(g,j) g=k when g>F, 2*(g-F)-1 <= k" and n is even;

T(n,k) = Sum_{j=2*(g-F)..k"} T(g,j) g=k when g>F, 2*(g-F) <= k" and n is odd.

Showing 1-4 of 4 results.

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A111133 Number of partitions of n into at least two distinct parts.

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A368484 Number of compositions (ordered partitions) of n into parts not greater than n/2.

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A368501 Number of compositions (ordered partitions) of n into distinct parts not greater than n/2.

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A262885 Irregular triangle T(n,k) read by rows: T(n,k) = number of partitions of n into at least two distinct parts, where the largest part is n-k.

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