A239482 -id:A239482 - cp's OEIS frontend

A008483 Number of partitions of n into parts >= 3.

1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

a(0) = 1 because the empty partition vacuously has each part >= 3. - Jason Kimberley, Jan 11 2011
Number of partitions where the largest part occurs at least three times. - Joerg Arndt, Apr 17 2011
By removing a single part of size 3, an A026796 partition of n becomes an A008483 partition of n - 3.
For n >= 3 the sequence counts the isomorphism classes of authentication codes AC(2,n,n) with perfect secrecy and with largest probability 0.5 that an interceptor could deceive with a substituted message. - E. Keith Lloyd (ekl(AT)soton.ac.uk).
For n >= 1, also the number of regular graphs of degree 2. - Mitch Harris, Jun 22 2005
(1 + 0*x + 0*x^2 + x^3 + x^4 + x^5 + 2*x^6 + ...) = (1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ...) * 1 / (1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + ...). - Gary W. Adamson, Jun 30 2009
Because the triangle A051031 is symmetric, a(n) is also the number of (n-3)-regular graphs on n vertices. Since the disconnected (n-3)-regular graph with minimum order is 2K_{n-2}, then for n > 4 there are no disconnected (n-3)-regular graphs on n vertices. Therefore for n > 4, a(n) is also the number of connected (n-3)-regular graphs on n vertices. - Jason Kimberley, Oct 05 2009
Number of partitions of n+2 such that 2*(number of parts) is a part. - Clark Kimberling, Feb 27 2014
For n >= 1, a(n) is the number of (1,1)-separable partitions of n, as defined at A239482.  For example, the (1,1)-separable partitions of 11 are [10,1], [7,1,2,1], [6,1,3,1], [5,1,4,1], [4,1,2,1,2,1], [3,1,3,1,2,1], so that a(11) = 6. - Clark Kimberling, Mar 21 2014
From Peter Bala, Dec 01 2024: (Start)
Let P(3, n) denote the set of partitions of n into parts k >= 3. Then A000041(n) = (1/2) * Sum_{parts k in all partitions in P(3, n+3)} phi(k), where phi(k) is the Euler totient function (see A000010). For example, with n = 5, there are 3 partitions of n + 3 = 8 into parts greater then 3, namely, 8, 5 + 3 and 4 + 4, and (1/2)*(phi(8) + phi(5) + phi(3) + 2*phi(4)) = 7 = A000041(5). (End)

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 301 terms from Vincenzo Librandi)
Peter Adams, Saad I. El-Zanati, Peter Florido, and William Turner, On 2-Factorizations of the Complete 3-Uniform Hypergraph of Order 12 Minus a 1-Factor, Combinatorics, Graph Theory and Computing (SEICCGTC 2021) Springer Proc. Math. Stat., Vol 448, pp. 383-392. See p. 326.
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
R.-Q. Feng, J. H. Kwak and E. K. Lloyd, Isomorphism classes of authentication codes, Bull. Austral. Math. Soc. 69 (2004), no. 2, 203-215.
Elisabeth Gaar and Daniel Krenn, Metamour-regular Polyamorous Relationships and Graphs, arXiv:2005.14121 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 446
F. Jouneau-Sion and O. Torres, In Fisher's net: exact F-tests in semi-parametric models with exchangeable errors, August 2014, preprint on ResearchGate.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g.
Johan Kok, Degree affinity number of certain 2-regular graphs, Open J. of Disc. Appl. Math. (2020) Vol. 3, No. 3, 77-84.
Eric Weisstein's World of Mathematics, Two-Regular Graph.

Essentially the same sequence as A026796 and A281356.
From Jason Kimberley, Nov 07 2009 and Jan 05 2011 and Feb 03 2011: (Start)
Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), this sequence (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).
2-regular simple graphs: A179184 (connected), A165652 (disconnected), this sequence (not necessarily connected).
2-regular not necessarily connected graphs without multiple edges [partitions without 2 as a part]: this sequence (no loops allowed [without 1 as a part]), A027336 (loops allowed [parts may be 1]).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), this sequence (g=3), A008484 (g=4), A185325 (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10), ... (End)
Cf. A008284.

p := NumberOfPartitions; A008483 :=  func< n | n eq 0 select 1 else n le 2 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3)>; // Jason Kimberley, Jan 11 2011

series(1/product((1-x^i),i=3..50),x,51);
ZL := [ B,{B=Set(Set(Z, card>=3))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..46); # Zerinvary Lajos, Mar 13 2007
with(combstruct):ZL2:=[S,{S=Set(Cycle(Z,card>2))}, unlabeled]:seq(count(ZL2,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
with(combstruct):a:=proc(m) [A,{A=Set(Cycle(Z,card>m))},unlabeled]; end: A008483:=a(2):seq(count(A008483,size=n),n=0..46); # Zerinvary Lajos, Oct 02 2007

f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 3], {n, 49}] (* Robert G. Wilson v, Jan 31 2011 *)
Rest[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Length[p]]], {n, 50}]]  (* Clark Kimberling, Feb 27 2014 *)

a(n) = numbpart(n)-numbpart(n-1)-numbpart(n-2)+numbpart(n-3) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import partition
def A008483(n): return partition(n)-partition(n-1)-partition(n-2)+partition(n-3) # Chai Wah Wu, Jun 10 2025

a(n) = p(n) - p(n - 1) - p(n - 2) + p(n - 3) where p(n) is the number of unrestricted partitions of n into positive parts (A000041).
G.f.: Product_{m>=3} 1/(1-x^m).
G.f.: (Sum_{n>=0} x^(3*n)) / (Product_{k=1..n} (1 - x^k)). - Joerg Arndt, Apr 17 2011
a(n) = A121081(n+3) - A121659(n+3). - Reinhard Zumkeller, Aug 14 2006
Euler transformation of A179184. a(n) = A179184(n) + A165652(n). - Jason Kimberley, Jan 05 2011
a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Feb 26 2015
G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018
a(n) = Sum_{j=0..floor(n/2)} A008284(n-2*j,j). - Gregory L. Simay, Apr 27 2023
G.f.: 1 + Sum_{n >= 1} x^(n+2)/Product_{k = 0..n-1} (1 - x^(k+3)). - Peter Bala, Dec 01 2024

A165652 Number of disconnected 2-regular graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 8, 9, 12, 16, 20, 24, 32, 38, 48, 59, 72, 87, 109, 129, 157, 190, 229, 272, 330, 390, 467, 555, 659, 778, 926, 1086, 1283, 1509, 1774, 2074, 2437, 2841, 3322, 3871, 4509, 5236, 6094, 7055, 8181, 9464, 10944, 12624, 14577, 16778, 19322, 22209

Offset: 0

Jason Kimberley, Sep 28 2009

a(n) is also the number of partitions of n such that each part i satisfies 2

For n>=2, it appears that a(n+1) is the number of (1,0)-separable partitions of n, as defined at A239482. For example, the four (1,0)-separable partitions of 9 are 621, 531, 441, 31212, corresponding to a(10) = 4. - Clark Kimberling, Mar 21 2014.

			The a(6)=1 graph is C_3+C_3. The a(7)=1 graph is C_3+C_4. The a(8)=2 graphs are C_3+C_5, C_4+C_4. The a(9)=3 graphs are 3C_3, C_3+C_6, C_4+C_5.

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

2-regular simple graphs: A179184 (connected), this sequence (disconnected), A008483 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A157928 (k=0), A157928 (k=1), this sequence (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8).
Disconnected 2-regular simple graphs with girth at least g: this sequence (g=3), A185224 (g=4), A185225 (g=5), A185226 (g=6), A185227 (g=7), A185228 (g=8), A185229 (g=9).
Cf. A239482.

p := NumberOfPartitions; a := func< n | n lt 3 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3) - 1 >;

a = A008483 - A179184 = Euler_tranformation(A179184) - A179184.
For n > 2, since there is exactly one connected 2-regular graph on n vertices (the n cycle C_n) then a(n) = A008483(n) - 1.
(A008483(n) is also the number of not necessarily connected 2-regular graphs on n vertices.)
Column D(n, 2) in the triangle A068933.

A239497 Number of partitions p of n such that if h = min(p), then h is an (h,1)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 7, 9, 11, 15, 17, 22, 28, 34, 40, 52, 60, 75, 90, 109, 129, 160, 186, 225, 268, 321, 376, 455, 530, 632, 743, 878, 1028, 1219, 1416, 1667, 1947, 2281, 2648, 3103, 3593, 4189, 4857, 5638, 6516, 7564, 8715, 10080, 11614, 13394, 15392

Offset: 1

Clark Kimberling, Mar 24 2014

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

			a(7) counts these partitions: 61, 52, 43, 3211.

Cf. A239498, A118096, A239500, A239501, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A239483 Number of (3,0)-separable partitions of n; see Comments.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 3, 4, 5, 7, 8, 10, 12, 16, 18, 22, 26, 33, 38, 45, 53, 65, 75, 89, 103, 124, 143, 168, 195, 230, 265, 309, 357, 418, 479, 556, 639, 742, 850, 979, 1122, 1294, 1478, 1696, 1935, 2220, 2528, 2889, 3287, 3752, 4261, 4850, 5502, 6257, 7084

Offset: 4

Clark Kimberling, Mar 20 2014

			The (3,0)-separable partitions of 11 are 731, 632, 434, 23231, so that a(11) = 4.

Cf. A239467, A165652, A239482, A239484, A239485.

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

A239484 Number of (4,0)-separable partitions of n; see Comments.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 19, 22, 26, 31, 36, 42, 51, 58, 68, 79, 92, 107, 125, 143, 165, 191, 221, 253, 293, 333, 383, 440, 503, 574, 657, 747, 853, 971, 1105, 1253, 1427, 1616, 1833, 2076, 2349, 2655, 3006, 3389, 3826, 4313, 4861

Offset: 5

Clark Kimberling, Mar 20 2014

			The (4,0)-separable partitions of 12 are 741, 642, 543, 24141, so that a(12) = 4.

Cf. A239467, A165652, A239482, A239483, A239485.

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

A239485 Number of (5,0)-separable partitions of n; see Comments.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 4, 4, 6, 7, 8, 9, 12, 13, 16, 19, 22, 25, 31, 34, 41, 47, 54, 62, 74, 82, 96, 110, 126, 143, 167, 187, 216, 245, 279, 316, 364, 408, 466, 527, 597, 673, 767, 860, 976, 1098, 1238, 1391, 1574, 1761, 1986, 2228, 2502, 2801, 3150, 3518

Offset: 6

Clark Kimberling, Mar 20 2014

			The (5,0)-separable partitions of 13 are 751, 652, 454, 15151, so that a(13) = 4.

Cf. A239467, A239481, A165652, A239483, A239484.

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

A239498 Number of partitions p of n such that if h = 2*min(p), then h is an (h,1)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 3, 1, 2, 5, 4, 4, 8, 7, 9, 15, 15, 18, 23, 26, 32, 43, 47, 57, 72, 80, 98, 120, 138, 163, 198, 227, 267, 323, 372, 438, 517, 596, 696, 818, 944, 1098, 1282, 1477, 1711, 1989, 2285, 2637, 3049, 3496, 4023, 4633, 5303, 6080, 6976, 7968

Offset: 1

Clark Kimberling, Mar 24 2014

			a(9) counts these partitions: 63, 4212, 212121.

Cf. A239497, A118096, A239500, A239501, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 29, 32, 36, 39, 43, 48, 53, 58, 65, 70, 78, 85, 93, 101, 112, 120, 132, 143, 156, 168, 184, 198, 216, 233, 253, 273, 298, 320, 348, 376, 407, 439

Offset: 1

Clark Kimberling, Mar 24 2014

			a(12) counts these partitions: 84, 4431, 4422.

Cf. A239497, A239498, A118096, A239501, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A239501 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,1)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 3, 3, 2, 2, 3, 5, 4, 8, 4, 5, 9, 6, 13, 10, 11, 15, 14, 17, 16, 20, 21, 26, 29, 30, 33, 36, 35, 41, 47, 47, 61, 61, 66, 71, 73, 85, 88, 98, 102, 114, 122, 131, 148, 154, 163, 182, 188, 205, 220, 231, 249, 271, 293, 306, 338, 359

Offset: 1

Clark Kimberling, Mar 24 2014

			a(11) counts these partitions: 4313, 4232, 321212.

Cf. A239497, A239498, A239499, A239500, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A239510 Number of partitions p of n such that if h = min(p), then h is an (h,0)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 18, 24, 30, 37, 48, 59, 73, 90, 109, 132, 163, 193, 233, 280, 334, 397, 475, 559, 663, 784, 924, 1085, 1279, 1494, 1751, 2049, 2392, 2784, 3248, 3769, 4382, 5081, 5887, 6808, 7879, 9087, 10486, 12083, 13910, 15988, 18384

Offset: 1

Clark Kimberling, Mar 24 2014

			a(9) counts these 5 partitions: 612, 513, 414, 423, 312121.

Cf. A239511, A237828, A239513, A239514, A239482.

z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)

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A008483 Number of partitions of n into parts >= 3.

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A165652 Number of disconnected 2-regular graphs on n vertices.

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A239497 Number of partitions p of n such that if h = min(p), then h is an (h,1)-separator of p; see Comments.

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A239483 Number of (3,0)-separable partitions of n; see Comments.

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A239484 Number of (4,0)-separable partitions of n; see Comments.

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A239485 Number of (5,0)-separable partitions of n; see Comments.

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A239498 Number of partitions p of n such that if h = 2*min(p), then h is an (h,1)-separator of p; see Comments.

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A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.

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