A182485 -id:A182485 - cp's OEIS frontend

A098859 Number of partitions of n into parts each of which is used a different number of times.

1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 28, 31, 45, 55, 62, 82, 105, 116, 153, 172, 208, 251, 312, 341, 431, 492, 588, 676, 826, 905, 1120, 1249, 1475, 1676, 2003, 2187, 2625, 2922, 3409, 3810, 4481, 4910, 5792, 6382, 7407, 8186, 9527, 10434

Offset: 0

David S. Newman, Oct 11 2004

Fill, Janson and Ward refer to these partitions as Wilf partitions. - Peter Luschny, Jun 04 2012

			a(6)=7 because 6= 4+1+1= 3+3= 3+1+1+1= 2+2+2= 2+1+1+1+1= 1+1+1+1+1+1. Four unrestricted partitions of 6 are not counted by a(6): 5+1, 4+2, 3+2+1 because at least two different summands are each used once; 2+2+1+1 because each summand is used twice.
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(1) = 1 through a(9) = 15 partitions are the following. The Heinz numbers of these partitions are given by A130091.
  1   2    3     4      5       6        7         8          9
      11   111   22     221     33       322       44         333
                 211    311     222      331       332        441
                 1111   2111    411      511       422        522
                        11111   3111     2221      611        711
                                21111    4111      2222       3222
                                111111   22111     5111       6111
                                         31111     22211      22221
                                         211111    41111      33111
                                         1111111   221111     51111
                                                   311111     411111
                                                   2111111    2211111
                                                   11111111   3111111
                                                              21111111
                                                              111111111
(End)

Simon Langowski and Mark Daniel Ward, Table of n, a(n) for n = 0..2000 (terms 0..300 from M. D. Ward, 301..700 from Maciej Ireneusz Wilczynski)
James Allen Fill, Svante Janson and Mark Daniel Ward, Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf, The Electronic Journal of Combinatorics, Volume 19, Issue 2 (2012)
Daniel Kane and Robert C. Rhoades, Asymptotics for Wilf's partitions with distinct multiplicities
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Simon Langowski, Program to compute Wilf Partitions, 2018
Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.
Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions; Local copy [Pdf file only, no active links]

Row sums of A182485.
Cf. A100471, A100881, A105637, A211858, A211859, A211860, A211861, A211862, A211863, A242882.
Cf. A047966 (each part appears the same number of times), A090858, A116608, A130091, A325242.

a098859 = p 0 [] 1 where
   p m ms _      0 = if m `elem` ms then 0 else 1
   p m ms k x
     | x < k       = 0
     | m == 0      = p 1 ms k (x - k) + p 0 ms (k + 1) x
     | m `elem` ms = p (m + 1) ms k (x - k)
     | otherwise   = p (m + 1) ms k (x - k) + p 0 (m : ms) (k + 1) x
-- Reinhard Zumkeller, Dec 27 2012

a[n_] := Length[sp = Split /@ IntegerPartitions[n]] - Count[sp, {__List, b_List, ___List, c_List, ___List} /; Length[b] == Length[c]]; Table[a[n], {n, 0, 48}] (* _Jean-François Alcover, Jan 17 2013 *)

a(n)={((r,k,b,w)->if(!k||!r, if(r,0,1), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b, 1<Andrew Howroyd, Aug 31 2019

log(a(n)) ~ N*log(N) where N = (6*n)^(1/3) (see Fill, Janson and Ward). - Peter Luschny, Jun 04 2012

Corrected and extended by Vladeta Jovovic, Oct 22 2004

A242896 Number T(n,k) of compositions of n into k parts with distinct multiplicities, where parts are counted without multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 0, 3, 3, 0, 2, 10, 0, 4, 12, 0, 2, 38, 0, 4, 56, 0, 3, 79, 0, 4, 152, 60, 0, 2, 251, 285, 0, 6, 284, 498, 0, 2, 594, 1438, 0, 4, 920, 2816, 0, 4, 1108, 5208, 0, 5, 2136, 11195, 0, 2, 3402, 24094, 0, 6, 4407, 38523, 0, 2, 8350, 85182

Offset: 0

Alois P. Heinz, May 25 2014

			T(5,1) = 2: [1,1,1,1,1], [5].
T(5,2) = 10: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 2;
  0, 2;
  0, 3,   3;
  0, 2,  10;
  0, 4,  12;
  0, 2,  38;
  0, 4,  56;
  0, 3,  79;
  0, 4, 152, 60;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-7 give: A000007, A000005, A242900, A246230, A246231, A246232, A246233, A246234.
Row sums give A242882.
Cf. A182485 (the same for partitions), A242887.

b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
      `if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, `if`(j=0, 1, x)*
       b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, {})):
seq(T(n), n=0..16);

b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n-i*j, i-1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

A182473 Number of partitions of n into exactly 2 different parts with distinct multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 8, 9, 12, 16, 22, 20, 31, 35, 34, 44, 51, 53, 62, 65, 68, 87, 86, 89, 95, 118, 108, 126, 127, 138, 142, 162, 154, 188, 160, 193, 189, 227, 204, 228, 221, 258, 238, 282, 247, 311, 272, 320, 284, 344, 318, 373, 327, 398, 334, 407, 380, 450

Offset: 0

Alois P. Heinz, Apr 30 2012

			a(4) = 1: [2,1,1], part 2 occurs once and part 1 occurs twice.
a(5) = 3: [2,1,1,1], [2,2,1], [3,1,1].
a(6) = 3: [2,1,1,1,1], [3,1,1,1], [4,1,1].
a(7) = 8: [2,1,1,1,1,1], [2,2,1,1,1], [2,2,2,1], [3,1,1,1,1], [3,2,2], [3,3,1], [4,1,1,1], [5,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..3000

Column k=2 of A182485.

Cf. A242900 (the same for compositions).

A212114 Number of partitions of n into exactly 3 different parts with distinct multiplicities.

Original entry on oeis.org

1, 4, 5, 12, 16, 24, 33, 52, 57, 89, 100, 131, 152, 207, 215, 292, 311, 381, 424, 527, 535, 679, 714, 826, 901, 1074, 1075, 1304, 1361, 1522, 1643, 1881, 1900, 2221, 2297, 2517, 2700, 3016, 3032, 3478, 3602, 3846, 4098, 4513, 4574, 5092, 5278, 5568, 5957, 6412

Offset: 10

Alois P. Heinz, May 01 2012

nonn

			a(10) = 1: [3,2,2,1,1,1].
a(11) = 4: [3,2,2,1,1,1,1], [3,2,2,2,1,1], [3,3,2,1,1,1], [4,2,2,1,1,1].
a(12) = 5: [3,2,2,1,1,1,1,1], [3,3,2,1,1,1,1], [4,2,2,1,1,1,1], [4,2,2,2,1,1], [5,2,2,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 10..170

Column k=3 of A182485.

A212115 Number of partitions of n into exactly 4 different parts with distinct multiplicities.

Original entry on oeis.org

1, 5, 8, 17, 29, 41, 59, 95, 120, 170, 224, 297, 367, 491, 583, 764, 904, 1118, 1306, 1631, 1863, 2269, 2576, 3099, 3479, 4179, 4583, 5456, 6029, 7041, 7693, 8978, 9745, 11274, 12174, 14037, 15074, 17202, 18383, 20978, 22404, 25276, 26825, 30271, 32092, 36051

Offset: 20

Alois P. Heinz, May 01 2012

nonn

			a(20) = 1: [4,3,3,2,2,2,1,1,1,1].
a(21) = 5: [4,3,3,2,2,2,1,1,1,1,1], [4,3,3,2,2,2,2,1,1,1], [4,3,3,3,2,2,1,1,1,1], [4,4,3,2,2,2,1,1,1,1], [5,3,3,2,2,2,1,1,1,1].
a(22) = 8: [4,3,3,2,2,2,1,1,1,1,1,1], [4,3,3,3,2,2,1,1,1,1,1], [4,4,3,2,2,2,1,1,1,1,1], [4,4,3,2,2,2,2,1,1,1], [5,3,3,2,2,2,1,1,1,1,1], [5,3,3,2,2,2,2,1,1,1], [5,3,3,3,2,2,1,1,1,1], [6,3,3,2,2,2,1,1,1,1].

Column k=4 of A182485.

A212116 Number of partitions of n into exactly 5 different parts with distinct multiplicities.

Original entry on oeis.org

1, 6, 12, 24, 48, 68, 108, 168, 232, 318, 458, 588, 781, 1043, 1298, 1663, 2147, 2603, 3216, 4039, 4814, 5822, 7103, 8356, 9936, 11902, 13784, 16160, 19082, 21841, 25301, 29399, 33450, 38226, 44084, 49543, 56381, 64047, 71870, 80596, 91459, 101249, 113545

Offset: 35

Alois P. Heinz, May 01 2012

nonn

			a(35) = 1: [5,4,4,3,3,3,2,2,2,2,1,1,1,1,1].
a(36) = 6: [5,4,4,3,3,3,2,2,2,2,1,1,1,1,1,1], [5,4,4,3,3,3,2,2,2,2,2,1,1,1,1], [5,4,4,3,3,3,3,2,2,2,1,1,1,1,1], [5,4,4,4,3,3,2,2,2,2,1,1,1,1,1], [5,5,4,3,3,3,2,2,2,2,1,1,1,1,1], [6,4,4,3,3,3,2,2,2,2,1,1,1,1,1].

Column k=5 of A182485.

A212117 Number of partitions of n into exactly 6 different parts with distinct multiplicities.

Original entry on oeis.org

1, 7, 17, 34, 74, 113, 186, 289, 434, 598, 888, 1162, 1610, 2115, 2783, 3534, 4716, 5756, 7364, 9125, 11404, 13721, 17206, 20391, 24983, 29767, 35781, 41987, 50733, 58665, 69642, 80987, 95040, 109085, 128243, 145889, 169528, 193419, 222437, 251663, 290151

Offset: 56

Alois P. Heinz, May 01 2012

nonn

			a(56) = 1: [6,5,5,4,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1].
a(57) = 7: [6,5,5,4,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1,1],[6,5,5,4,4,4,3,3,3,3,2,2,2,2,2,2,1,1,1,1,1],[6,5,5,4,4,4,3,3,3,3,3,2,2,2,2,1,1,1,1,1,1],
  [6,5,5,4,4,4,4,3,3,3,2,2,2,2,2,1,1,1,1,1,1], [6,5,5,5,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1],
  [6,6,5,4,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1],[7,5,5,4,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1].

Column k=6 of A182485.

A212118 Number of partitions of n into exactly 7 different parts with distinct multiplicities.

Original entry on oeis.org

1, 8, 23, 48, 109, 184, 306, 496, 774, 1096, 1655, 2272, 3175, 4264, 5794, 7424, 9992, 12585, 16255, 20354, 25979, 31619, 39971, 48283, 59782, 71799, 87907, 104004, 126743, 148911, 178680, 209259, 249489, 288937, 342755, 395090, 464036

Offset: 84

Alois P. Heinz, May 01 2012

nonn

			a(84) = 1: [7,6,6,5,5,5,4,4,4,4,3,3,3,3,3,2,2,2,2,2,2,1,1,1,1,1,1,1].

Column k=7 of A182485.

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A098859 Number of partitions of n into parts each of which is used a different number of times.

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A242896 Number T(n,k) of compositions of n into k parts with distinct multiplicities, where parts are counted without multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.

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A182473 Number of partitions of n into exactly 2 different parts with distinct multiplicities.

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A212114 Number of partitions of n into exactly 3 different parts with distinct multiplicities.

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A212115 Number of partitions of n into exactly 4 different parts with distinct multiplicities.

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A212116 Number of partitions of n into exactly 5 different parts with distinct multiplicities.

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A212117 Number of partitions of n into exactly 6 different parts with distinct multiplicities.

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A212118 Number of partitions of n into exactly 7 different parts with distinct multiplicities.

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