A323433 -id:A323433 - cp's OEIS frontend

A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

1, 1, 2, 3, 6, 10, 16, 26, 42, 64, 100, 150, 224, 330, 482, 697, 999, 1418, 1996, 2794, 3879, 5355, 7343, 10018, 13583, 18338, 24618, 32917, 43790, 58043, 76591, 100716, 131906, 172194, 223966, 290423, 375318, 483668, 621368, 796138, 1017146

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

If a partition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of partitions.

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(1)  (21)(21)  (31)(31)   (41)(41)
                         (111)(1)  (22)(2)    (32)(32)
                                   (211)(11)  (311)(11)
                                   (211)(21)  (311)(31)
                                   (1111)(1)  (221)(21)
                                              (221)(22)
                                              (2111)(11)
                                              (2111)(21)
                                              (11111)(1)

With multiplicity we have A339006.
The version for compositions is A355384.
The homogeneous version w/o containment is A355385, compositions A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.
Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,IntegerPartitions[n]}],{n,0,15}]

A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 20, 24, 31, 38, 48, 59, 72, 86, 106, 125, 150, 180, 213, 250, 296, 347, 407, 477, 555, 645, 751, 869, 1003, 1161, 1334, 1534, 1763, 2018, 2306, 2637, 3002, 3418, 3886, 4409, 4994, 5659, 6390, 7214, 8135, 9160, 10300, 11580, 12990

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(10) = 20 split partitions:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [4 3]
  [1] [2] [3] [4] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Alois P. Heinz, Table of n, a(n) for n = 0..7000

Cf. A000005, A000219, A101509, A117433, A316245, A319066, A319794, A323295, A323301, A323431, A323433.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, numtheory[tau](t), b(n, i-1, t)+
         b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n>i(i+1)/2, 0,
     If[n == 0, DivisorSigma[0, t], b[n, i-1, t] +
     b[n-i, Min[n-i, i-1], t+1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

a(n) = Sum_y A000005(k), where the sum is over all strict integer partitions of n and k is the number of parts.

A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 0, 4, 0, 0, 1, 12, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 12, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

			The a(60) = 12 matrices:
  [2 2] [2 2] [2 3] [2 3] [2 5] [2 5] [3 2] [3 2] [3 5] [5 2] [5 2] [5 3]
  [3 5] [5 3] [2 5] [5 2] [2 3] [3 2] [2 5] [5 2] [2 2] [2 3] [3 2] [2 2]

Positions of 0's are A323521.
Positions of 1's are A323520.
Cf. A000290, A008480, A026478, A056239, A089299, A103198, A112798, A120732, A323433, A323525, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[IntegerQ[Sqrt[PrimeOmega[n]]],Length[Permutations[primeMS[n]]],0],{n,100}]

If A001222(n) is a perfect square, then a(n) = A008480(n). Otherwise, a(n) = 0.

A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 17, 22, 26, 35, 40, 51, 60, 75, 86, 109, 124, 153, 175, 214, 243, 297, 336, 403, 456, 546, 614, 731, 821, 975, 1095, 1283, 1437, 1689, 1887, 2195, 2448, 2851, 3172, 3676, 4083, 4724, 5245, 6022, 6677, 7695, 8504, 9720

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(9) = 9 splittings:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)    (5,2)    (6,2)    (6,3)
                                 (3,2,1)  (6,1)    (7,1)    (7,2)
                                          (4,2,1)  (4,3,1)  (8,1)
                                                   (5,2,1)  (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)
                                                            (4),(3,2)
The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal sums is A318683.
The version with strictly decreasing sums is A318684.
The version with weakly decreasing sums is A319794.
The version with different sums is A336132.
Starting with a composition gives A304961.
Starting with a non-strict partition gives A336134.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336127, A336128, A336130, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 31, 40, 73, 98, 158, 204, 340, 420, 629, 819, 1202, 1494, 2174, 2665, 3759, 4688, 6349, 7806, 10788, 13035, 17244, 21128, 27750, 33499, 43941, 52627, 67957, 81773, 103658, 124047, 158628, 187788, 235162, 280188, 349612, 413120, 513952, 604568

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(5) = 15 splittings:
  (1)  (2)      (3)          (4)              (5)
       (1,1)    (2,1)        (2,2)            (3,2)
       (1),(1)  (1,1,1)      (3,1)            (4,1)
                (1),(1,1)    (2,1,1)          (2,2,1)
                (1),(1),(1)  (2),(2)          (3,1,1)
                             (1,1,1,1)        (2,1,1,1)
                             (2),(1,1)        (2),(2,1)
                             (1),(1,1,1)      (1,1,1,1,1)
                             (1,1),(1,1)      (2),(1,1,1)
                             (1),(1),(1,1)    (1),(1,1,1,1)
                             (1),(1),(1),(1)  (1,1),(1,1,1)
                                              (1),(1),(1,1,1)
                                              (1),(1,1),(1,1)
                                              (1),(1),(1),(1,1)
                                              (1),(1),(1),(1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..60

The version with weakly decreasing sums is A316245.
The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with strictly decreasing sums is A336135.
The version with different sums is A336131.
Starting with a composition gives A075900.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A304961, A305551, A318684, A323433, A336128, A336130, A336133.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],LessEqual@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r >= t && t >= s, self()(r,m,t,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m),s,t+m,1))); recurse(n,n,0,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 5, 3, 1, 1, 0, 7, 3, 2, 1, 1, 0, 11, 6, 4, 2, 1, 1, 0, 15, 7, 5, 3, 2, 1, 1, 0, 22, 12, 7, 6, 3, 2, 1, 1, 0, 30, 14, 11, 7, 5, 3, 2, 1, 1, 0, 42, 22, 14, 11, 8, 5, 3, 2, 1, 1, 0, 56, 27, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 77, 40, 27, 21, 15, 12, 7, 5, 3, 2, 1, 1

Offset: 0

Seiichi Manyama, May 14 2023

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  1,  1;
  0,  5,  3,  1, 1;
  0,  7,  3,  2, 1, 1;
  0, 11,  6,  4, 2, 1, 1;
  0, 15,  7,  5, 3, 2, 1, 1;
  0, 22, 12,  7, 6, 3, 2, 1, 1;
  0, 30, 14, 11, 7, 5, 3, 2, 1, 1;
  ...

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

Row sums give A323433.
Column k=0..5 give A000007, A000041, A027187, A363045, A363046, A363047.
T(2n,n) gives A052810.
Cf. A008284, A072233, A350890.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), add(
    (j-> b(n-j, min(n-j, j)))(k*i), i=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, May 14 2023

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[j, b[n - j, Min[n - j, j]]][k*i], {i, 0, n/k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)

T(n, k) = sum(j=0, n, #partitions(n-k*j, k*j));

For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i)/Product_{j=1..k*i} (1-x^j).

A355389 Number of unordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 10, 21, 55, 105, 231, 435, 861, 1540, 2926, 5050, 9045, 15400, 26565, 43956, 73920, 119805, 196251, 313236, 501501, 786885, 1239525, 1915903, 2965830, 4528545, 6909903, 10417330, 15699606, 23403061, 34848726, 51435153, 75761895, 110744403, 161577276

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

			The a(0) = 0 through a(4) = 10 pairs:
  .  .  (2)(11)  (3)(21)    (4)(22)
                 (3)(111)   (4)(31)
                 (21)(111)  (22)(31)
                            (4)(211)
                            (22)(211)
                            (31)(211)
                            (4)(1111)
                            (22)(1111)
                            (31)(1111)
                            (211)(1111)

The version for compositions is A006516.
Without distinctness we get A086737.
The unordered version is A355390, without distinctness A001255.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319794, A319910, A323433, A339006, A355385.

a:= n-> binomial(combinat[numbpart](n),2):
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 07 2024

Table[Binomial[PartitionsP[n],2],{n,0,6}]

a(n) = binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

a(n) = binomial(A000041(n), 2) = A355390(n)/2.

A355390 Number of ordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 2, 6, 20, 42, 110, 210, 462, 870, 1722, 3080, 5852, 10100, 18090, 30800, 53130, 87912, 147840, 239610, 392502, 626472, 1003002, 1573770, 2479050, 3831806, 5931660, 9057090, 13819806, 20834660, 31399212, 46806122, 69697452, 102870306, 151523790, 221488806

Offset: 0

Gus Wiseman, Jul 04 2022

nonn

			The a(0) = 0 through a(3) = 6 pairs:
  .  .  (11)(2)  (21)(3)
        (2)(11)  (3)(21)
                 (111)(3)
                 (3)(111)
                 (111)(21)
                 (21)(111)

Without distinctness we have A001255, unordered A086737.
The version for compositions is A020522, unordered A006516.
The unordered version is A355389.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319910, A323433, A323583, A339006, A355385.

Table[Length[Select[Tuples[IntegerPartitions[n],2],UnsameQ@@#&]],{n,0,15}]

a(n) = 2*binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

a(n) = 2*A355389(n) = 2*binomial(A000041(n), 2).

A373030 Expansion of 1 + Sum_{i>=1} Sum_{j>=1} x^(ij) Product_{k=1..i*j-1} (1+x^k).

Original entry on oeis.org

1, 1, 2, 4, 5, 7, 11, 14, 17, 24, 30, 37, 48, 58, 71, 92, 108, 129, 160, 188, 225, 273, 319, 377, 449, 524, 612, 721, 836, 969, 1134, 1305, 1503, 1742, 1996, 2291, 2637, 3008, 3435, 3929, 4469, 5076, 5778, 6541, 7401, 8393, 9466, 10676, 12049, 13550, 15235, 17128

Offset: 0

Seiichi Manyama, May 20 2024

nonn

Row sums of A373029.

Cf. A000005, A000009, A323433.

G.f.: 1 + Sum_{k>=1} A000005(k) * x^k * Product_{j=1..k-1} (1+x^j).

cp's OEIS Frontend

A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

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A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

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Formula

A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

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A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

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Mathematica

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

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Extensions

A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

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A355389 Number of unordered pairs of distinct integer partitions of n.

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Maple

Mathematica

PARI

Formula

A355390 Number of ordered pairs of distinct integer partitions of n.

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Author

Keywords

Examples

Crossrefs

Programs

A373030 Expansion of 1 + Sum_{i>=1} Sum_{j>=1} x^(ij) Product_{k=1..i*j-1} (1+x^k).