A266499 -id:A266499 - cp's OEIS frontend

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).

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

Emeric Deutsch and Alois P. Heinz, Dec 29 2015

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).

			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;

Alois P. Heinz, Rows n = 0..50, flattened

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.

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);

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 *)

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

Gus Wiseman, May 17 2022

nonn

			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).

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.

Table[Length[Select[IntegerPartitions[n], Times@@#==Times@@Length/@Split[#]&]],{n,0,30}]

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

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

A353698 Number of integer partitions of n whose product equals their length.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 3, 0, 3, 2, 2, 1, 5, 0, 1, 2, 5, 1, 4, 0, 3, 3, 2, 1, 4, 2, 3, 2, 2, 0, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 5, 1, 4, 0, 4, 3, 3, 1, 6, 2, 3, 4, 4, 2, 4, 1, 4, 2, 3, 1, 8, 2, 4, 2, 4, 2, 5, 2, 4, 2

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(n) partitions for selected n (A..H = 10..17):
n=9:    n=21:             n=27:                 n=33:
---------------------------------------------------------------------------
51111   B1111111111       E1111111111111        H1111111111111111
321111  72111111111111    921111111111111111    B211111111111111111111
        531111111111111   54111111111111111111  831111111111111111111111
        4221111111111111                        5511111111111111111111111
                                                333111111111111111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..10000

The LHS (product of parts) is counted by A339095, rank statistic A003963.
The RHS (length) is counted by A008284, rank statistic A001222.
These partitions are ranked by A353699.
A266477 counts partitions by product of multiplicities, rank stat A005361.
A353504 counts partitions w/ product less than product of multiplicities.
A353505 counts partitions w/ product greater than product of multiplicities.
A353506 counts partitions w/ prod equal to prod of mults, ranked by A353503.
Cf. A000041, A002033, A098859, A114640, A181819, A225485, A266499, A319000, A325280, A353398, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#==Length[#]&]],{n,0,30}]

a(r,m=r,p=1,k=0) = {(p==k+r) + sum(m=2, min(m, (k+r)\p),  self()(r-m, min(m,r-m), p*m, k+1))} \\ Andrew Howroyd, Jan 02 2023

Terms a(61) and beyond from Andrew Howroyd, Jan 02 2023

A353741 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with product k, all zeros removed.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 1, 1, 4, 2, 2, 1, 4, 1, 1, 1, 3, 2

Offset: 0

Gus Wiseman, May 20 2022

Warning: There are certain internal "holes" in A339095 that are removed in this sequence.

			Triangle begins:
  1
  1
  1 1
  1 1 1
  1 1 1 2
  1 1 1 2 1 1
  1 1 1 2 1 2 2 1
  1 1 1 2 1 2 1 2 1 1 2
  1 1 1 2 1 2 1 3 1 1 3 1 3 1
  1 1 1 2 1 2 1 3 2 1 3 1 1 3 2 2 2 1
  1 1 1 2 1 2 1 3 2 2 3 1 1 4 2 2 1 4 1 1 1 3 2
Row n = 7 counts the following partitions:
  1111111   211111   31111   4111    511   61     7   421    331   52   43
                             22111         3211       2221              322

Row sums are A000041.
Row lengths are A034891.
A partial transpose is A319000.
The full version with zeros is A339095, rank statistic A003963.
A008284 counts partitions by sum, strict A116608.
A225485 counts partitions by frequency depth.
A266477 counts partitions by product of multiplicities, ranked by A005361.
Cf. A002033, A266499, A325242, A325268, A325280, A353503, A353506, A353698.

DeleteCases[Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,10},{k,1,2^n}],0,2]

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A353698 Number of integer partitions of n whose product equals their length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A353741 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with product k, all zeros removed.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica