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.

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A266477 Triangle read by rows in which T(n,k) is the number of partitions of n with product of multiplicities of parts equal to k; n>=0, 1<=k<=A266480(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 2, 2, 0, 1, 5, 4, 2, 1, 1, 1, 1, 6, 6, 2, 3, 1, 2, 0, 2, 8, 7, 4, 4, 1, 2, 1, 0, 2, 1, 10, 8, 6, 6, 3, 2, 1, 3, 0, 1, 0, 2, 12, 13, 6, 6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1, 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0
Offset: 0

Views

Author

Emeric Deutsch and Alois P. Heinz, Dec 29 2015

Keywords

Comments

Sum of entries in row n = A000041(n) = number of partitions of n.
T(n,1) = A000009(n) = number of partitions of n into distinct parts.
T(n,2) = A090858(n).
T(n,3) = A265251(n).
Smallest row m >= 0 with T(m,n) > 0 is A266325(n).
T(n,A266480(n)) gives A266871(n).

Examples

			Row 4 is [2,2,0,1]. Indeed, the products of the multiplicities of the parts in the partitions [4], [1,3], [2,2], [1,1,2], [1,1,1,1] are 1, 1, 2, 2, 4, respectively.
Triangle T(n,k) begins:
00 :  1;
01 :  1;
02 :  1,  1;
03 :  2,  0, 1;
04 :  2,  2, 0,  1;
05 :  3,  2, 1,  0, 1;
06 :  4,  2, 2,  2, 0, 1;
07 :  5,  4, 2,  1, 1, 1, 1;
08 :  6,  6, 2,  3, 1, 2, 0, 2;
09 :  8,  7, 4,  4, 1, 2, 1, 0, 2, 1;
10 : 10,  8, 6,  6, 3, 2, 1, 3, 0, 1, 0, 2;
11 : 12, 13, 6,  6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1;
12 : 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0, 2, 0, 0, 0, 2, 0, 1;

Links

Crossrefs

Columns k=1-10 give: A000009, A090858, A265251, A266687, A266688, A266689, A266690, A266691, A266692, A266693.
Main diagonal gives A266499.
Row lengths give A266480.
Cf. A000041, A077285, A266325, A266871.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      x^max(p, p*n), add(b(n-i*j, i-1, max(p, p*j)), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..16);
  • Mathematica
    Map[Table[Length@ Position[#, k], {k, Max@ #}] &, #] &@ Table[Map[Times @@ Map[Last, Tally@ #] &, IntegerPartitions@ n], {n, 12}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, x^Max[p, p*n], Sum[b[n - i*j, i - 1, Max[p, p*j]], {j, 0, n/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p,x]}]][ b[n, n, 1]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Formula

Sum_{k>=1} k*T(n,k) = A077285(n).
G.f. of column p if p is prime: Sum_{k>0} x^(p*k)/(1+x^k) * Product_{i>0} (1+x^i), giving the number of partitions of n such that there is exactly one part which occurs p times, while all other parts occur only once.
If p is prime then column p is asymptotic to 3^(1/4) * c(p) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)), where c(p) = Sum_{j>=0} (-1)^j/(j+p) = (PolyGamma((p+1)/2) - PolyGamma(p/2))/2. - Vaclav Kotesovec, May 24 2018

A353506 Number of integer partitions of n whose parts have the same product as their multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 3, 3, 2, 3, 2, 0, 2, 3, 2, 1, 3, 1, 6, 3, 2, 3, 3, 2, 3, 4, 1, 2, 3, 6, 3, 2, 2, 3, 3, 1, 2, 6, 6, 4, 7, 2, 3, 6, 4, 3, 3, 0, 4, 5, 3, 5, 5, 6, 5, 3, 3, 3, 6, 5, 5, 6, 6, 3, 3, 3, 4, 4, 4, 6, 7, 2, 5, 7, 6, 2, 3, 4, 6, 11, 9, 4, 4, 1, 5, 6, 4, 7, 9, 6, 4
Offset: 0

Views

Author

Gus Wiseman, May 17 2022

Keywords

Examples

			The a(0) = 1 through a(18) = 2 partitions:
  n= 0: ()
  n= 1: (1)
  n= 2:
  n= 3:
  n= 4: (211)
  n= 5:
  n= 6: (3111) (2211)
  n= 7:
  n= 8: (41111)
  n= 9:
  n=10: (511111)
  n=11: (32111111)
  n=12: (6111111) (22221111)
  n=13: (322111111)
  n=14: (71111111) (4211111111)
  n=15:
  n=16: (811111111) (4411111111) (42211111111)
  n=17: (521111111111) (332111111111) (322211111111)
  n=18: (9111111111) (333111111111)
For example, the partition y = (322111111) has multiplicities (1,2,6) with product 12, and the product of parts is also 3*2*2*1*1*1*1*1*1 = 12, so y is counted under a(13).

Crossrefs

LHS (product of parts) is ranked by A003963, counted by A339095.
RHS (product of multiplicities) is ranked by A005361, counted by A266477.
For shadows instead of prime exponents we have A008619, ranked by A003586.
Taking sum instead of product of parts gives A266499.
For shadows instead of prime indices we have A353398, ranked by A353399.
These partitions are ranked by A353503.
Taking sum instead of product of multiplicities gives A353698.
A008284 counts partitions by length.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A353507 gives product of multiplicities (of exponents) in prime signature.
Cf. A085629, A114640, A116608, A118914, A124010, A319000, A325702, A353394, A353500.
Cf. A000792, A266480.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Times@@#==Times@@Length/@Split[#]&]],{n,0,30}]
  • PARI
    a(n) = {my(nb=0); forpart(p=n, my(s=Set(p), v=Vec(p)); if (vecprod(vector(#s, i, #select(x->(x==s[i]), v))) == vecprod(v), nb++);); nb;} \\ Michel Marcus, May 20 2022

Extensions

a(71)-a(100) from Alois P. Heinz, May 20 2022

A266871 Number of partitions of n that maximize the product of multiplicities of parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1
Offset: 0

Views

Author

Emeric Deutsch and Alois P. Heinz, Jan 04 2016

Keywords

Examples

			a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2] (product of multiplicities = 8).
a(9) = 1: [1,1,1,1,1,2,2] (product = 10).
a(10) = 2: [1,1,1,1,1,1,2,2], [1,1,1,1,2,2,2] (product = 12).
a(11) = 1: [1,1,1,1,1,2,2,2] (product = 15).
a(23) = 3: [1,1,1,1,1,1,1,1,1,2,2,2,2,3,3], [1,1,1,1,1,1,1,1,2,2,2,3,3,3], [1,1,1,1,1,1,2,2,2,2,3,3,3] (product = 72).

Links

Crossrefs

Cf. A266477, A266480.

Programs

  • Maple
    b:= proc(n, i) option remember; local r,l,j;
      if n=0 or i=1 then [max(1, n),1]
    else r:= b(n, i-1);
         for j to iquo(n, i) do
           l:= (w-> [w[1]*j, w[2]])(b(n-i*j, i-1));
           r:= `if`(l[1]>r[1], l,
               `if`(l[1]=r[1], [0, l[2]], 0)+r)
         od; r
      fi
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..120);
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{r, l, j}, If[n == 0 || i == 1, {Max[1, n], 1}, r = b[n, i - 1]; For[j = 1, j <= Quotient[n, i], j++, l = Function[w, {w[[1]]*j, w[[2]]}][b[n - i*j, i - 1]]; r = If[l[[1]] > r[[1]], l, If[l[[1]] == r[[1]], {0, l[[2]]}, 0] + r]]; r]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)

Formula

a(n) = A266477(n,A266480(n)).
Showing 1-3 of 3 results.

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