A183568 -id:A183568 - cp's OEIS frontend

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

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

A184639 Number of partitions of n having no parts with multiplicity 4.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 14, 19, 27, 37, 50, 67, 88, 115, 153, 196, 253, 324, 412, 524, 661, 828, 1036, 1290, 1603, 1980, 2443, 2997, 3671, 4487, 5460, 6631, 8034, 9703, 11703, 14075, 16890, 20226, 24175, 28838, 34332, 40801, 48394, 57307, 67765, 79974

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(4) = 4, because 4 partitions of 4 have no parts with multiplicity 4: [1,1,2], [2,2], [1,3], [4].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183561, A183568, A007690, A116645, A118807, A184640, A184641, A184642, A184643, A184644, A184645.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=4, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[l, If[j == 4, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183561(n).
a(n) = A183568(n,0) - A183568(n,4).
G.f.: Product_{j>0} (1-x^(4*j)+x^(5*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-4*x) + exp(-6*x)) dx = 0.77101366877372033648945034346499691865027592089088481444183... - Vaclav Kotesovec, Jun 12 2025

A184640 Number of partitions of n having no parts with multiplicity 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 14, 21, 28, 39, 51, 72, 92, 124, 160, 210, 266, 349, 438, 562, 704, 892, 1107, 1395, 1720, 2141, 2631, 3249, 3965, 4873, 5916, 7216, 8730, 10585, 12742, 15387, 18443, 22151, 26466, 31646, 37659, 44873, 53212, 63149, 74666, 88295

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(5) = 6, because 6 partitions of 5 have no parts with multiplicity 5: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183562, A183568, A007690, A116645, A118807, A184639, A184641, A184642, A184643, A184644, A184645.

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[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 5, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183562(n).
a(n) = A183568(n,0) - A183568(n,5).
G.f.: Product_{j>0} (1-x^(5*j)+x^(6*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-5*x) + exp(-7*x)) dx = 0.78834765570757713777493985857868631321765157344539753651545... - Vaclav Kotesovec, Jun 12 2025

A184641 Number of partitions of n having no parts with multiplicity 6.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 29, 40, 54, 72, 96, 127, 166, 216, 279, 358, 457, 580, 735, 924, 1159, 1446, 1799, 2228, 2752, 3388, 4158, 5087, 6207, 7551, 9165, 11093, 13401, 16144, 19412, 23286, 27882, 33310, 39727, 47289, 56191, 66647, 78923, 93299

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(6) = 10, because 10 partitions of 6 have no parts with multiplicity 6: [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2,3], [3,3], [1,1,4], [2,4], [1,5], [6].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183563, A183568, A007690, A116645, A118807, A184639, A184640, A184642, A184643, A184644, A184645.

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[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 6, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183563(n).
a(n) = A183568(n,0) - A183568(n,6).
G.f.: Product_{j>0} (1-x^(6*j)+x^(7*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-6*x) + exp(-8*x)) dx = 0.79818518024793359047735154473665146019665210453617381247423... - Vaclav Kotesovec, Jun 12 2025

A184642 Number of partitions of n having no parts with multiplicity 7.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 22, 29, 41, 54, 75, 97, 130, 168, 222, 283, 368, 465, 597, 750, 949, 1183, 1488, 1841, 2292, 2822, 3487, 4267, 5239, 6376, 7782, 9429, 11439, 13798, 16661, 20007, 24043, 28763, 34420, 41021, 48894, 58066, 68956, 81627, 96592

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(7) = 14, because 14 partitions of 7 have no parts with multiplicity 7: [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,1,3], [1,1,2,3], [2,2,3], [1,3,3], [1,1,1,4], [1,2,4], [3,4], [1,1,5], [2,5], [1,6], [7].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183564, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184643, A184644, A184645.

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[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 7, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(FreeQ[Length/@Split[#],7]&)],{n,0,50}] (* _Harvey P. Dale, Sep 21 2024 *)

a(n) = A000041(n) - A183564(n).
a(n) = A183568(n,0) - A183568(n,7).
G.f.: Product_{j>0} (1-x^(7*j)+x^(8*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-7*x) + exp(-9*x)) dx = 0.80430417180776436899064351977235191494130305607975798117531... - Vaclav Kotesovec, Jun 12 2025

A184643 Number of partitions of n having no parts with multiplicity 8.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 30, 41, 55, 75, 99, 131, 172, 223, 288, 372, 474, 603, 764, 962, 1206, 1509, 1876, 2326, 2878, 3543, 4351, 5330, 6506, 7921, 9623, 11655, 14085, 16987, 20434, 24529, 29392, 35138, 41930, 49947, 59381, 70474, 83512, 98779

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183565, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184644, A184645.

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[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 8, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183565(n).
a(n) = A183568(n,0) - A183568(n,8).
G.f.: Product_{j>0} (1-x^(8*j)+x^(9*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-8*x) + exp(-10*x)) dx = 0.80836901097063952622501649557292291036896118821761722817375... - Vaclav Kotesovec, Jun 12 2025

A184644 Number of partitions of n having no parts with multiplicity 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 42, 55, 76, 99, 133, 172, 227, 290, 376, 477, 612, 769, 975, 1217, 1528, 1895, 2359, 2907, 3592, 4400, 5403, 6584, 8034, 9742, 11823, 14272, 17234, 20713, 24897, 29803, 35674, 42542, 50719, 60272, 71592, 84794

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183566, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184645.

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[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 9, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183566(n).
a(n) = A183568(n,0) - A183568(n,9).
G.f.: Product_{j>0} (1-x^(9*j)+x^(10*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-9*x) + exp(-11*x)) dx = 0.81120660452677002313966407107688916817839171627473737415672... - Vaclav Kotesovec, Jun 12 2025

cp's OEIS Frontend

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

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

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

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

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A184639 Number of partitions of n having no parts with multiplicity 4.

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A184640 Number of partitions of n having no parts with multiplicity 5.

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A184641 Number of partitions of n having no parts with multiplicity 6.

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