A317449 -id:A317449 - cp's OEIS frontend

A035310 Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

1, 4, 12, 47, 170, 750, 3255, 16010, 81199, 448156, 2579626, 15913058, 102488024, 698976419, 4976098729, 37195337408, 289517846210, 2352125666883, 19841666995265, 173888579505200, 1577888354510786, 14820132616197925, 143746389756336173, 1438846957477988926

Offset: 1

Alford Arnold

Ways of partitioning an n-multiset with multiplicities some partition of n.

Number of multiset partitions of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. The (weakly) normal version is A255906. - Gus Wiseman, Dec 31 2019

			a(3) = 12 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=3, f(12)=4, f(30)=5 and 3+4+5 = 12.
From _Gus Wiseman_, Dec 31 2019: (Start)
The a(1) = 1 through a(3) = 12 multiset partitions of strongly normal multisets:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A025487, A000041, A000110, A035098, A080688.
Sequence A035341 counts the ordered cases. Tables A093936 and A095705 distribute the values; e.g. 81199 = 30 + 536 + 3036 + 6181 + 10726 + 11913 + 14548 + 13082 + 21147.
Cf. A035341, A093936, A095705.
Row sums of A317449.
The uniform case is A317584.
The case with empty intersection is A317755.
The strict case is A317775.
The constant case is A047968.
The set-system case is A318402.
The case of strict parts is A330783.
Multiset partitions of integer partitions are A001970.
Unlabeled multiset partitions are A007716.
Cf. A001055, A255906, A269134, A316652, A317654, A318284, A330467, A330471, A330475, A330675.

with(numtheory):
g:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
b:= proc(n, i, l)
      `if`(n=0, g(mul(ithprime(t)^l[t], t=1..nops(l))$2),
      `if`(i<1, 0, add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> b(n$2, []):
seq(a(n), n=1..10);  # Alois P. Heinz, May 26 2013

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; b[n_, i_, l_] := If[n == 0, g[p = Product[Prime[t]^l[[t]], {t, 1, Length[l]}], p], If[i < 1, 0, Sum[b[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := b[n, n, {}]; Table[Print[an = a[n]]; an, {n, 1, 13}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020

from sympy.core.cache import cacheit
from sympy import divisors, isprime, prime
from operator import mul
@cacheit
def g(n, k):
    return (0 if n > k else 1) + (0 if isprime(n) else sum(g(n//d, d) for d in divisors(n)[1:-1] if d <= k))
@cacheit
def b(n, i, l):
    if n==0:
        p = reduce(mul, (prime(t + 1)**l[t] for t in range(len(l))))
        return g(p, p)
    else:
        return 0 if i<1 else sum([b(n - i*j, i - 1, l + [i]*j) for j in range(n//i + 1)])
def a(n):
    return b(n, n, [])
for n in range(1, 11): print(a(n)) # Indranil Ghosh, Aug 19 2017, after Maple code

More terms from Erich Friedman.
81199 from Alford Arnold, Mar 04 2008
a(10) from Alford Arnold, Mar 31 2008
a(10) corrected by Alford Arnold, Aug 07 2008
a(11)-a(13) from Alois P. Heinz, May 26 2013
a(14) from Alois P. Heinz, Sep 27 2014
a(15) from Alois P. Heinz, Jan 10 2015
Terms a(16) and beyond from Andrew Howroyd, Dec 30 2020

A055884 Euler transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 4, 5, 1, 4, 8, 7, 7, 1, 6, 12, 16, 12, 11, 1, 6, 17, 25, 28, 19, 15, 1, 8, 22, 43, 49, 48, 30, 22, 1, 8, 30, 58, 87, 88, 77, 45, 30, 1, 10, 36, 87, 134, 167, 151, 122, 67, 42, 1, 10, 45, 113, 207, 270, 296, 247, 185, 97, 56, 1, 12, 54, 155, 295, 448, 510, 507, 394, 278, 139, 77

Offset: 1

Christian G. Bower, Jun 09 2000

Number of multiset partitions of length-k integer partitions of n. - Gus Wiseman, Nov 09 2018

			From _Gus Wiseman_, Nov 09 2018: (Start)
Triangle begins:
   1
   1   2
   1   2   3
   1   4   4   5
   1   4   8   7   7
   1   6  12  16  12  11
   1   6  17  25  28  19  15
   1   8  22  43  49  48  30  22
   1   8  30  58  87  88  77  45  30
   ...
The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
  {{5}}   {{1,4}}     {{1,1,3}}       {{1,1,1,2}}         {{1,1,1,1,1}}
          {{2,3}}     {{1,2,2}}      {{1},{1,1,2}}       {{1},{1,1,1,1}}
         {{1},{4}}   {{1},{1,3}}     {{1,1},{1,2}}       {{1,1},{1,1,1}}
         {{2},{3}}   {{1},{2,2}}     {{2},{1,1,1}}      {{1},{1},{1,1,1}}
                     {{2},{1,2}}    {{1},{1},{1,2}}     {{1},{1,1},{1,1}}
                     {{3},{1,1}}    {{1},{2},{1,1}}    {{1},{1},{1},{1,1}}
                    {{1},{1},{3}}  {{1},{1},{1},{2}}  {{1},{1},{1},{1},{1}}
                    {{1},{2},{2}}
(End)

Alois P. Heinz, Rows n = 1..200, flattened
N. J. A. Sloane, Transforms

Row sums give A001970.
Main diagonal gives A000041.
Columns k=1-2 give: A057427, A052928.
T(n+2,n+1) gives A000070.
T(2n,n) gives A360468.
Cf. A055885, A055886, A360742.
Cf. A000219, A007716, A008284, A255906, A317449, A317532, A317533, A320796, A320801, A320808.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 17 2023

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* Gus Wiseman, Nov 09 2018 *)

A321449 Regular triangle read by rows where T(n,k) is the number of twice-partitions of n with a combined total of k parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 4, 5, 5, 0, 1, 4, 8, 8, 7, 0, 1, 6, 13, 19, 16, 11, 0, 1, 6, 17, 27, 32, 24, 15, 0, 1, 8, 24, 47, 61, 62, 41, 22, 0, 1, 8, 30, 63, 99, 111, 100, 61, 30, 0, 1, 10, 38, 94, 158, 209, 210, 170, 95, 42, 0, 1, 10, 45, 119, 229, 328, 382, 348, 259, 136, 56

Offset: 0

Gus Wiseman, Nov 10 2018

A twice partition of n (A063834) is a choice of an integer partition of each part in an integer partition of n.

			Triangle begins:
   1
   0   1
   0   1   2
   0   1   2   3
   0   1   4   5   5
   0   1   4   8   8   7
   0   1   6  13  19  16  11
   0   1   6  17  27  32  24  15
   0   1   8  24  47  61  62  41  22
   0   1   8  30  63  99 111 100  61  30
The sixth row {0, 1, 6, 13, 19, 16, 11} counts the following twice-partitions:
  (6)  (33)    (222)      (2211)        (21111)          (111111)
       (42)    (321)      (3111)        (1111)(2)        (111)(111)
       (51)    (411)      (111)(3)      (111)(21)        (1111)(11)
       (3)(3)  (21)(3)    (211)(2)      (21)(111)        (11111)(1)
       (4)(2)  (22)(2)    (21)(21)      (211)(11)        (11)(11)(11)
       (5)(1)  (31)(2)    (22)(11)      (2111)(1)        (111)(11)(1)
               (3)(21)    (221)(1)      (11)(11)(2)      (1111)(1)(1)
               (32)(1)    (3)(111)      (111)(2)(1)      (11)(11)(1)(1)
               (4)(11)    (31)(11)      (11)(2)(11)      (111)(1)(1)(1)
               (41)(1)    (311)(1)      (2)(11)(11)      (11)(1)(1)(1)(1)
               (2)(2)(2)  (11)(2)(2)    (21)(11)(1)      (1)(1)(1)(1)(1)(1)
               (3)(2)(1)  (2)(11)(2)    (211)(1)(1)
               (4)(1)(1)  (21)(2)(1)    (11)(2)(1)(1)
                          (2)(2)(11)    (2)(11)(1)(1)
                          (22)(1)(1)    (21)(1)(1)(1)
                          (3)(11)(1)    (2)(1)(1)(1)(1)
                          (31)(1)(1)
                          (2)(2)(1)(1)
                          (3)(1)(1)(1)

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

Row sums are A063834. Last column is A000041.

Cf. A000219, A001970, A007716, A008284, A055884, A289501, A317449, A317532, A317533, A320801, A320808.

g:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      g(n, i-1)+ `if`(i>n, 0, expand(g(n-i, i)*x)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, i-1)+ `if`(i>n, 0, expand(b(n-i, i)*g(i$2))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 11 2018

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],Length[Join@@#]==k&],{ptn,IntegerPartitions[n]}]],{n,0,10},{k,0,n}]
(* Second program: *)
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, x^n,
     g[n, i - 1] + If[i > n, 0, Expand[g[n - i, i]*x]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n,
     b[n, i - 1] + If[i > n, 0, Expand[b[n - i, i]*g[i, i]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

O.g.f.: Product_{n >= 0} 1/(1 - x^n * (Sum_{0 <= k <= n} A008284(n,k) * t^k)).

A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.

Original entry on oeis.org

1, 1, 3, 13, 59, 313, 1847, 11977, 84483, 642405, 5228987, 45297249, 415582335, 4021374193, 40895428051, 435721370413, 4850551866619, 56282199807401, 679220819360775, 8508809310177481, 110454586096508563, 1483423600240661781, 20581786429087269819

Offset: 0

Gus Wiseman, Aug 06 2018

nonn

			The a(3) = 13 strict multiset partitions:
  {{1,1,1}}, {{1},{1,1}},
  {{1,2,2}}, {{1},{2,2}}, {{2},{1,2}},
  {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
  {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A001055, A007716, A045778, A255906, A281116, A317449, A317532, A317583, A317653, A317752, A317757, A317775.

Row sums of A327116.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 16 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@#&]],{n,9}]
(* Second program: *)
c := Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {k, 0, n}, {i, 0, k}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

a(0), a(8)-a(22) from Alois P. Heinz, Sep 16 2019

A317775 Number of strict multiset partitions of strongly normal multisets of size n, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 3, 10, 36, 136, 596, 2656, 13187, 68226, 381572, 2233091, 13940407, 90981030, 626911429, 4509031955, 33987610040, 266668955183, 2180991690286, 18512572760155, 163103174973092, 1487228204311039, 14027782824491946, 136585814043190619, 1371822048393658001, 14190528438090988629

Offset: 1

Gus Wiseman, Aug 06 2018

nonn

			The a(3) = 10 strict multiset partitions:
  {{1,1,1}}, {{1},{1,1}},
  {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
  {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A001055, A007716, A045778, A281116, A317449, A317584, A317654, A317755, A317757, A317776.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],UnsameQ@@#&]],{n,6}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1,-n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p,n))); s[n]+=1; s/2} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

cp's OEIS Frontend

A035310 Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

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A055884 Euler transform of partition triangle A008284.

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Maple

Mathematica

A321449 Regular triangle read by rows where T(n,k) is the number of twice-partitions of n with a combined total of k parts.

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Comments

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Programs

Maple

Mathematica

Formula

A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A317775 Number of strict multiset partitions of strongly normal multisets of size n, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions