cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204
Offset: 0

Views

Author

Wouter Meeussen, Jul 09 2005

Keywords

Comments

Counted as orderless pairs since intersection is commutative.

Examples

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Links

Crossrefs

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

Programs

  • Mathematica
    using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)
  • PARI
    a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

Formula

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017
a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Extensions

Name edited by Gus Wiseman, Oct 10 2023
a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A258281 Number of partitions of 3 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 4, 5, 13, 18, 37, 56, 103, 154, 279, 398, 682, 1027, 1664, 2433, 3977, 5755, 8957, 13173, 19980, 29002, 43894, 62562, 92531, 133550, 193348, 274049, 398218, 558839, 796906, 1120833, 1577874, 2197279, 3089063, 4258348, 5915878, 8170572, 11231601, 15355764
Offset: 5

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Examples

			a(7) = 4: [7;6,1;5,2], [7;6,1;4,3], [7;5,2;4,3], [6,1;5,2;4,3].

Links

Crossrefs

Column k=3 of A258280.

Programs

  • Mathematica
    nmax = 30; p = 1; Do[p = Expand[p*(1 + x^j + y^j + z^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) && (Exponent[#, z] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] == Exponent[#, z] &]; Table[Coefficient[p, x^n*y^n*z^n]/6, {n, 5, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)

Formula

a(n) = 1/6 * [(x*y*z)^n] Product_{j>0} (1+x^j+y^j+z^j).

A258282 Number of partitions of 4 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 5, 7, 20, 31, 75, 118, 266, 401, 841, 1345, 2581, 4023, 7758, 11932, 21543, 33941, 59099, 91816, 159059, 242955, 407523, 634827, 1032202, 1576276, 2572429, 3888631, 6177754, 9400070, 14665453, 22055721, 34349618, 51033150, 78166821, 116800290, 176090572
Offset: 7

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Examples

			a(9) = 5: [9;8,1;7,2;6,3], [9;8,1;7,2;5,4], [9;8,1;6,3;5,4], [9;7,2;6,3;5,4], [8,1;7,2;6,3;5,4].

Links

Crossrefs

Column k=4 of A258280.

Formula

a(n) = 1/4! * [(x*y*z*u)^n] Product_{j>0} (1+x^j+y^j+z^j+u^j).

A258283 Number of partitions of 5 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 6, 8, 29, 47, 134, 213, 559, 939, 2178, 3564, 8184, 13225, 27904, 46376, 93151, 153022, 302948, 489271, 931845, 1540667, 2818014, 4565768, 8369128, 13423638, 23782711, 38552461, 66714197, 106889113, 184313769, 291358130, 491972639, 785299762, 1298230358
Offset: 9

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Examples

			a(11) = 6: [11;10,1;9,2;8,3;7,4], [11;10,1;9,2;8,3;6,5], [11;10,1;9,2;7,4;6,5], [11;10,1;8,3;7,4;6,5], [11;9,2;8,3;7,4;6,5], [10,1;9,2;8,3;7,4;6,5].

Links

Crossrefs

Column k=5 of A258280.

Formula

a(n) = 1/5! * [(x*y*z*u*v)^n] Product_{j>0} (1+x^j+y^j+z^j+u^j+v^j).

A258284 Number of partitions of 6 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 7, 10, 40, 64, 215, 373, 1076, 1823, 5052, 8519, 21279, 36836, 86354, 148550, 338354, 571638, 1242227, 2156661, 4453444, 7573979, 15612927, 26325001, 52042210, 88853970, 170752721, 288367364, 550431942, 916369374, 1706148831, 2877961566, 5219075925
Offset: 11

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Examples

			a(13) = 7: [13;12,1;11,2;10,3;9,4;8,5], [13;12,1;11,2;10,3;9,4;7,6], [13;12,1;11,2;10,3;8,5;7,6], [13;12,1;11,2;9,4;8,5;7,6], [13;12,1;10,3;9,4;8,5;7,6], [13;11,2;10,3;9,4;8,5;7,6], [12,1;11,2;10,3;9,4;8,5;7,6].

Links

Crossrefs

Column k=6 of A258280.

Formula

a(n) = 1/6! * [(x*y*z*u*v*w)^n] Product_{j>0} (1+x^j+y^j+z^j+u^j+v^j+w^j).

A258285 Number of partitions of 7 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 8, 11, 52, 91, 326, 567, 1925, 3361, 10114, 17998, 50084, 89303, 236472, 414418, 1036205, 1872828, 4394636, 7781405, 18111119, 31868623, 70513669, 125728510, 269084037, 475275290, 1005278492, 1749029589, 3594394017, 6355048530, 12651552078, 22036692845, 43804062866, 75652893831
Offset: 13

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Links

Crossrefs

Column k=7 of A258280.

Formula

a(n) = 1/7! * [Product_{i=1..7} x_i^n] Product_{j>0} (1+Sum_{i=1..7} x_i^j).

Extensions

More terms from Alois P. Heinz, Apr 03 2017

A258286 Number of partitions of 8 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 9, 13, 66, 111, 475, 852, 3128, 5668, 19011, 34804, 108328, 195267, 567674, 1059291, 2849122, 5212695, 13785287, 25156313, 62613272, 115799409, 277245609, 508970856, 1196118973, 2161689541
Offset: 15

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Crossrefs

Column k=8 of A258280.

Formula

a(n) = 1/8! * [Product_{i=1..8} x_i^n] Product_{j>0} (1+Sum_{i=1..8} x_i^j).

A258287 Number of partitions of 9 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 10, 14, 82, 145, 652, 1198, 4880, 9095, 33810, 61933, 212087, 405116, 1261067, 2366558, 7163718, 13477035, 37969720, 72363916, 194919282
Offset: 17

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Crossrefs

Column k=9 of A258280.

Formula

a(n) = 1/9! * [Product_{i=1..9} x_i^n] Product_{j>0} (1+Sum_{i=1..9} x_i^j).

A258288 Number of partitions of 10 copies of n into distinct parts.

Original entry on oeis.org

1, 1, 11, 16, 99, 174, 875, 1657, 7352, 13655, 55787, 107946, 393317, 751343, 2624771, 5056414, 16243829, 31688464, 96658985
Offset: 19

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Crossrefs

Column k=10 of A258280.

Formula

a(n) = 1/10! * [Product_{i=1..10} x_i^n] Product_{j>0} (1+Sum_{i=1..10} x_i^j).

A258289 Number of partitions of 1, 2, 3, or more copies of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 9, 17, 21, 43, 57, 109, 157, 301, 447, 895, 1307, 2663, 4207, 8463, 13283, 28489, 45151, 95485, 157767, 336711, 561603, 1236963, 2061173, 4567227, 7946575, 17516101, 30324977, 69519697, 121465499, 276609723, 496333307, 1137900605
Offset: 0

Views

Author

Alois P. Heinz, May 25 2015

Keywords

Examples

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [2].
a(3) = 3: [3], [2,1], [3;2,1].
a(4) = 3: [4], [3,1], [4;3,1].
a(5) = 7: [5], [4,1], [3,2], [5;4,1], [5;3,2], [4,1;3,2], [5;4,1;3,2].
a(7) = 17: [7], [6,1], [5,2], [4,3], [4,2,1], [7;6,1], [7;5,2], [7;4,3], [7;4,2,1], [6,1;5,2], [6,1;4,3], [5,2;4,3], [7;6,1;5,2], [7;6,1;4,3], [7;5,2;4,3], [6,1;5,2;4,3], [7;6,1;5,2;4,3].

Crossrefs

Cf. A000009, A065033, A258280.

Programs

  • Maple
    b:= proc() option remember; local m; m:= args[nargs];
     `if`(nargs=1, 1, `if`(args[1]=0, b(args[t] $t=2..nargs),
     `if`(m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2, 0,
      b(args[t] $t=1..nargs-1, m-1)+add(`if`(args[j]-m<0, 0,
      b(sort([seq(args[i]-`if`(i=j, m, 0), i=1..nargs-1)])[]
      , m-1)), j=1..nargs-1))))
    end:
a:= n-> add(b(n$k+1)/k!, k=1..max(1, ceil(n/2))):
seq(a(n), n=0..20);
  • Mathematica
    disParts[n_] := disParts[n] = Select[IntegerPartitions[n], Length[#] == Length[Union[#]]&];
T[n_, k_] := Select[Subsets[disParts[n], {k}], Length[Flatten[#]] == Length[Union[Flatten[#]]]&] // Length;
a[n_] := a[n] = If[n == 0, 1, Sum[T[n, k], {k, 1, Quotient[n+1, 2]}]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, May 01 2022 *)

Formula

a(n) = Sum_{k=1..A065033(n)} A258280(n,k).
a(n) = Sum_{k=1..max(1,ceiling(n/2))} 1/k! * [Product_{i=1..k} x_i^n] Product_{j>0} (1+Sum_{i=1..k} x_i^j).
Showing 1-10 of 12 results. Next

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