A316979 -id:A316979 - cp's OEIS frontend

A316980 Number of non-isomorphic strict multiset partitions of weight n.

1, 1, 3, 8, 23, 63, 197, 588, 1892, 6140, 20734, 71472, 254090, 923900, 3446572, 13149295, 51316445, 204556612, 832467052, 3455533022, 14621598811, 63023667027, 276559371189, 1234802595648, 5606647482646, 25875459311317, 121324797470067, 577692044073205

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(1)(1) <-> (123)
  (1)(2)(2) <-> (1)(23)
  (1)(2)(3) <-> (1)(2)(3)

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091.
Cf. A316974, A316978, A316979, A316981, A316983.
Cf. A049311, A059201, A319558, A319560, A319567.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(p=sum(t=1, n, subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*polcoef(exp(p-subst(p,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 21 2023

Euler transform of A319557. - Gus Wiseman, Sep 23 2018

a(7)-a(10) from Gus Wiseman, Sep 23 2018

Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023

A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 9, 17, 36, 72, 155, 319, 677, 1429, 3094, 6648, 14518, 31796, 70491, 156818, 352371, 795952, 1813580, 4155367, 9594425, 22283566, 52122379, 122631874, 290432439, 691831161, 1658270316, 3997272089, 9692519896, 23631827354, 57943821449, 142834652193

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n, under row and column permutations.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity.

			Non-isomorphic representatives of the a(4) = 9 self-dual multiset partitions:
  (1111),
  (1)(222), (2)(122), (11)(22), (12)(12),
  (1)(1)(23), (1)(2)(33), (1)(3)(23),
  (1)(2)(3)(4).
The a(4) = 9 square symmetric matrices:
. [4]
.
. [3 0]  [2 0]  [2 1]  [1 1]
. [0 1]  [0 2]  [1 0]  [1 1]
.
. [2 0 0]  [1 1 0]  [0 1 1]
. [0 1 0]  [1 0 0]  [1 0 0]
. [0 0 1]  [0 0 1]  [1 0 0]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A320796.
Main diagonal of A318805.
Cf. A000009, A001055, A007716, A007717, A020555, A045778.
Cf. A316974, A316978, A316979, A316980, A316981.

vector(25, n, n--; T(n,n)) \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

Terms a(9) and beyond from Andrew Howroyd, Sep 03 2018

A319564 Number of T_0 integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 21, 29, 40, 53, 73, 95, 128, 168, 221, 282, 368, 466, 599, 759, 962, 1201, 1513, 1881, 2345, 2901, 3590, 4407, 5416, 6614, 8083, 9827, 11937, 14442, 17458, 21021, 25299, 30347, 36363, 43438, 51843, 61705, 73384, 87054, 103149, 121949

Offset: 0

Gus Wiseman, Sep 23 2018

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks).

Also the number of integer partitions of n with no equivalent primes. In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent. See A316978 for more examples.

Cf. A000009, A000041, A001970, A007716, A059201, A305148, A316978, A316979, A316983, A319558, A319616, A319728.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}]
Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&]],{n,20}]

A316978 Number of factorizations of n into factors > 1 with no equivalent primes.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 5, 1, 4, 1, 4, 1, 1, 1, 7, 2, 1, 3, 4, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 4, 4, 1, 1, 12, 2, 4, 1, 4, 1, 7, 1, 7, 1, 1, 1, 7, 1, 1, 4, 11, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 1, 12, 5, 1, 1, 7, 1, 1

Offset: 1

Gus Wiseman, Jul 18 2018

nonn

In a factorization, two primes are equivalent if each factor has in its prime factorization the same multiplicity of both primes.

			The a(36) = 7 factorizations are (2*2*3*3), (2*2*9), (2*3*6), (3*3*4), (2*18), (3*12), (4*9). Missing from this list are (6*6) and (36).

Cf. A001055, A007716, A007717, A020555, A045778, A162247, A281116, A303386.

Cf. A316974, A316979, A316980, A316981.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[facs[n],UnsameQ@@dual[primeMS/@#]&]],{n,100}]

a(prime^n) = A000041(n).

a(squarefree) = 1.

A322847 Numbers whose prime indices have no equivalent primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

The complement is {13, 26, 29, 43, 47, 52, 58, 73, 79, 86, 94, ...}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent while {2,5} and {3,5} are not. In (30,6) also, the primes {2,3} are equivalent, while {2,5} and {3,5} are not.
Also MM-numbers of T_0 multiset multisystems. A multiset multisystem is a finite multiset of finite multisets. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated parts).

			The prime indices of 339 are {2, 30}, in which the primes {3,5} are equivalent, so 339 is not in the sequence.

Cf. A056239, A059201, A112798, A302242, A316978, A316979, A316983, A319558, A319564, A319728, A322846.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Select[Range[100],UnsameQ@@dual[primeMS/@primeMS[#]]&]

A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

Original entry on oeis.org

1, 1, 2, 6, 15, 40, 121

Offset: 0

Gus Wiseman, Jul 18 2018

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows and no equal columns.

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 6 strict multiset partitions with no equivalent vertices (first column) and their duals (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(2)(3) <-> (1)(2)(3)

Cf. A000009, A001055, A007716, A007717, A020555, A045778.

Cf. A316974, A316978, A316979, A316980, A316983.

A322846 Squarefree numbers whose prime indices have no equivalent primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 51, 53, 55, 57, 59, 61, 62, 65, 66, 67, 69, 70, 71, 74, 77, 78, 82, 83, 85, 87, 89, 91, 93, 95, 97, 102, 103, 105, 106, 107, 109, 110, 111, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent while {2,5} and {3,5} are not. In (30,6) also, the primes {2,3} are equivalent, while {2,5} and {3,5} are not.
Also MM-numbers of strict T_0 multiset multisystems. A multiset multisystem is a finite multiset of finite multisets. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated parts).

			The sequence of all strict T_0 multiset multisystems together with their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}
  34: {{},{4}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  38: {{},{1,1,1}}
  39: {{1},{1,2}}

Cf. A000009, A005117, A056239, A059201, A112798, A302242, A302505, A316978, A316979, A316983, A319558, A319564, A319728, A322847.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Select[Range[100],And[SquareFreeQ[#],UnsameQ@@dual[primeMS/@primeMS[#]]]&]

cp's OEIS Frontend

A316980 Number of non-isomorphic strict multiset partitions of weight n.

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A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

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A319564 Number of T_0 integer partitions of n.

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A316978 Number of factorizations of n into factors > 1 with no equivalent primes.

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A322847 Numbers whose prime indices have no equivalent primes.

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A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

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A322846 Squarefree numbers whose prime indices have no equivalent primes.

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