A183561 -id:A183561 - cp's OEIS frontend

A183558 Number of partitions of n containing a clique of size 1.

0, 1, 1, 2, 3, 6, 7, 13, 16, 25, 33, 49, 61, 90, 113, 156, 198, 269, 334, 448, 556, 726, 902, 1163, 1428, 1827, 2237, 2817, 3443, 4302, 5219, 6478, 7833, 9632, 11616, 14197, 17031, 20712, 24769, 29925, 35688, 42920, 50980, 61059, 72318, 86206, 101837, 120941

Offset: 0

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
  (1)  (2)  (3)   (4)    (5)     (6)      (7)       (8)
            (21)  (31)   (32)    (42)     (43)      (53)
                  (211)  (41)    (51)     (52)      (62)
                         (221)   (321)    (61)      (71)
                         (311)   (411)    (322)     (332)
                         (2111)  (3111)   (331)     (422)
                                 (21111)  (421)     (431)
                                          (511)     (521)
                                          (2221)    (611)
                                          (3211)    (3221)
                                          (4111)    (4211)
                                          (31111)   (5111)
                                          (211111)  (32111)
                                                    (41111)
                                                    (311111)
                                                    (2111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Column k=1 of A183568.
Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
Cf. A052485, A090858, A117571, A127002, A325241, A325242, A325244.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);

max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Oct 01 2014. Edited by Gus Wiseman, Apr 19 2019 *)

G.f.: (1-Product_{j>0} (1-x^(j)+x^(2*j))) / (Product_{j>0} (1-x^j)).
From Vaclav Kotesovec, Nov 15 2016: (Start)
a(n) = A000041(n) - A007690(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)

a(0)=0 prepended by Gus Wiseman, Apr 19 2019

A183568 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n containing a clique of size k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 0, 1, 5, 3, 2, 0, 1, 7, 6, 2, 1, 0, 1, 11, 7, 3, 2, 1, 0, 1, 15, 13, 5, 3, 1, 1, 0, 1, 22, 16, 9, 3, 3, 1, 1, 0, 1, 30, 25, 10, 6, 3, 2, 1, 1, 0, 1, 42, 33, 16, 8, 5, 3, 2, 1, 1, 0, 1, 56, 49, 23, 13, 6, 5, 2, 2, 1, 1, 0, 1, 77, 61, 31, 15, 10, 5, 5, 2, 2, 1, 1, 0, 1

Offset: 0

Alois P. Heinz, Jan 05 2011

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique. Each partition has a clique of size 0.

			T(5,2) = 2, because 2 (of 7) partitions of 5 contain (at least) one clique of size 2: [1,2,2], [1,1,3].
Triangle T(n,k) begins:
   1;
   1,  1;
   2,  1, 1;
   3,  2, 0, 1;
   5,  3, 2, 0, 1;
   7,  6, 2, 1, 0, 1;
  11,  7, 3, 2, 1, 0, 1;
  15, 13, 5, 3, 1, 1, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
Differences between columns 0 and k (0A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184644, A184645.
T(2*k+1,k+1) gives A002865.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=k, [l[1]$2], l))(b(n-i*j, i-1, k)), j=0..n/i)))
    end:
T:= (n, k)-> (l-> l[`if`(k=0, 1, 2)])(b(n, n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == k, {l[[1]], l[[1]]}, l]][b[n - i*j, i-1, k]], {j, 0, n/i}]] ]; t[n_, k_] := Function[l, l[[If[k == 0, 1, 2]]]][b[n, n, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

G.f. of column k: (1-Product_{j>0} (1-x^(k*j)+x^((k+1)*j))) / (Product_{j>0} (1-x^j)).

A183563 Number of partitions of n containing a clique of size 6.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 8, 10, 15, 18, 27, 33, 47, 57, 78, 96, 129, 159, 208, 258, 330, 407, 517, 635, 798, 978, 1217, 1482, 1833, 2225, 2729, 3303, 4028, 4856, 5885, 7070, 8528, 10211, 12259, 14628, 17494, 20800, 24777, 29378, 34867

Offset: 6

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(10) = 2, because 2 partitions of 10 contain (at least) one clique of size 6: [1,1,1,1,1,1,2,2], [1,1,1,1,1,1,4].

Alois P. Heinz, Table of n, a(n) for n = 6..1000

6th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=6, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=6..55);

max = 55; f = (1 - Product[1 - x^(6j) + x^(7j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 6] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(6*j)+x^(7*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184641(n). - Vaclav Kotesovec, Jun 12 2025

A183559 Number of partitions of n containing a clique of size 2.

Original entry on oeis.org

1, 0, 2, 2, 3, 5, 9, 10, 16, 23, 31, 43, 60, 75, 106, 140, 179, 237, 310, 389, 508, 647, 815, 1032, 1305, 1617, 2033, 2527, 3117, 3857, 4764, 5812, 7142, 8711, 10585, 12866, 15605, 18803, 22716, 27325, 32774, 39286, 47016, 56019, 66819, 79456, 94273, 111766

Offset: 2

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(7) = 5, because 5 partitions of 7 contain (at least) one clique of size 2: [1,1,1,2,2], [1,1,2,3], [2,2,3], [1,3,3], [1,1,5].

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A183568.

Cf. A000041, A183558, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=2, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=2..50);

max = 50; f = (1 - Product[1 - x^(2j) + x^(3j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 2] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(2*j)+x^(3*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A116645(n). - Vaclav Kotesovec, Jun 12 2025

A183560 Number of partitions of n containing a clique of size 3.

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 6, 8, 13, 15, 24, 30, 44, 54, 77, 98, 134, 165, 222, 279, 367, 454, 588, 731, 936, 1148, 1454, 1788, 2241, 2732, 3400, 4140, 5106, 6183, 7579, 9157, 11156, 13406, 16249, 19482, 23489, 28042, 33666, 40087, 47914, 56851

Offset: 3

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(9) = 6, because 6 partitions of 9 contain (at least) one clique of size 3: [1,1,1,2,2,2], [2,2,2,3], [1,1,1,3,3], [3,3,3], [1,1,1,2,4], [1,1,1,6].

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A183568.

Cf. A000041, A183558, A183559, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=3, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=3..50);

max = 50; f = (1 - Product[1 - x^(3j) + x^(4j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 3] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(3*j)+x^(4*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A118807(n). - Vaclav Kotesovec, Jun 12 2025

A183562 Number of partitions of n containing a clique of size 5.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 5, 9, 11, 16, 21, 31, 36, 52, 65, 88, 110, 148, 180, 238, 295, 379, 469, 600, 731, 926, 1133, 1413, 1725, 2141, 2590, 3194, 3864, 4719, 5692, 6924, 8301, 10049, 12026, 14468, 17263, 20694, 24586, 29359, 34804, 41372

Offset: 5

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(11) = 5, because 5 partitions of 11 contain (at least) one clique of size 5: [1,1,1,1,1,2,2,2], [1,2,2,2,2,2], [1,1,1,1,1,3,3], [1,1,1,1,1,2,4], [1,1,1,1,1,6].

Alois P. Heinz, Table of n, a(n) for n = 5..1000

5th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=5, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=5..55);

max = 55; f = (1 - Product[1 - x^(5j) + x^(6j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 5] (* Jean-François Alcover, Oct 01 2014 *)
Table[Count[IntegerPartitions[n,{5,PartitionsP[n]}],?(MemberQ[ Length/@ Split[ #],5]&)],{n,5,60}] (* _Harvey P. Dale, Feb 02 2019 *)

G.f.: (1-Product_{j>0} (1-x^(5*j)+x^(6*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184640(n). - Vaclav Kotesovec, Jun 12 2025

A183564 Number of partitions of n containing a clique of size 7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 17, 25, 30, 42, 53, 72, 87, 117, 144, 188, 231, 298, 365, 466, 567, 714, 871, 1085, 1316, 1630, 1972, 2422, 2918, 3562, 4280, 5195, 6219, 7507, 8966, 10773, 12815, 15335, 18196, 21680, 25653, 30453

Offset: 7

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(13) = 4, because 4 partitions of 13 contain (at least) one clique of size 7: [1,1,1,1,1,1,1,2,2,2], [1,1,1,1,1,1,1,3,3], [1,1,1,1,1,1,1,2,4], [1,1,1,1,1,1,1,6].

Alois P. Heinz, Table of n, a(n) for n = 7..1000

7th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=7, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=7..55);

max = 55; f = (1 - Product[1 - x^(7j) + x^(8j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 7] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(7*j)+x^(8*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184642(n). - Vaclav Kotesovec, Jun 12 2025

A183565 Number of partitions of n containing a clique of size 8.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 8, 9, 13, 16, 24, 28, 40, 49, 66, 82, 110, 132, 175, 214, 274, 336, 428, 520, 655, 798, 990, 1203, 1486, 1793, 2200, 2653, 3227, 3880, 4701, 5622, 6779, 8092, 9701, 11546, 13793, 16355, 19466, 23029, 27290, 32199, 38048, 44752, 52719

Offset: 8

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(12) = 2, because 2 partitions of 12 contain (at least) one clique of size 8: [1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,4].

Alois P. Heinz, Table of n, a(n) for n = 8..1000

8th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=8, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=8..60);

max = 60; f = (1 - Product[1 - x^(8j) + x^(9j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 8] (* Jean-François Alcover, Oct 01 2014 *)
c8[n_]:=If[MemberQ[Tally[n][[All,2]],8],1,0]; Table[Total[c8/@ IntegerPartitions[ x]],{x,8,60}] (* Harvey P. Dale, Aug 12 2018 *)

G.f.: (1-Product_{j>0} (1-x^(8*j)+x^(9*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184643(n). - Vaclav Kotesovec, Jun 12 2025

A183566 Number of partitions of n containing a clique of size 9.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 9, 13, 15, 23, 27, 38, 47, 63, 77, 103, 126, 165, 201, 258, 315, 401, 487, 611, 743, 924, 1118, 1382, 1664, 2041, 2455, 2989, 3583, 4340, 5185, 6248, 7446, 8930, 10604, 12668, 15002, 17848, 21083, 24987, 29435, 34776, 40860

Offset: 9

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(12) = 1, because 1 partition of 12 contains (at least) one clique of size 9: [1,1,1,1,1,1,1,1,1,3].

Alois P. Heinz, Table of n, a(n) for n = 9..1000

9th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=9, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=9..60);

max=60;f=(1-Product[1-x^(9j)+x^(10j),{j,1,max}])/Product[1-x^j,{j,1,max}]; s=Series[f,{x,0,max}]; Drop[CoefficientList[s,x],9] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(9*j)+x^(10*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184644(n). - Vaclav Kotesovec, Jun 12 2025

A183567 Number of partitions of n containing a clique of size 10.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 15, 22, 26, 37, 45, 61, 74, 99, 120, 157, 192, 247, 299, 381, 462, 580, 703, 874, 1055, 1303, 1569, 1921, 2309, 2808, 3363, 4070, 4859, 5848, 6964, 8342, 9903, 11817, 13988, 16623, 19626, 23240, 27363, 32297

Offset: 10

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(14) = 2, because 2 partitions of 14 contain (at least) one clique of size 10: [1,1,1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,1,1,4].

Alois P. Heinz, Table of n, a(n) for n = 10..1000

10th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=10..60);

max = 60; f = (1 - Product[1 - x^(10j) + x^(11j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 10] (* Jean-François Alcover, Oct 01 2014 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Length/@Split[#],10]&]],{n,10,60}] (* Harvey P. Dale, Oct 02 2021 *)

G.f.: (1-Product_{j>0} (1-x^(10*j)+x^(11*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184645(n). - Vaclav Kotesovec, Jun 12 2025

cp's OEIS Frontend

A183558 Number of partitions of n containing a clique of size 1.

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A183568 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n containing a clique of size k.

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A183563 Number of partitions of n containing a clique of size 6.

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A183559 Number of partitions of n containing a clique of size 2.

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A183560 Number of partitions of n containing a clique of size 3.

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A183562 Number of partitions of n containing a clique of size 5.

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A183564 Number of partitions of n containing a clique of size 7.

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