A091357 -id:A091357 - cp's OEIS frontend

A089924 Number of plane partitions of n with 3 or more columns.

0, 0, 1, 3, 8, 19, 41, 85, 167, 319, 588, 1066, 1880, 3272, 5588, 9431, 15686, 25841, 42070, 67926, 108634, 172467, 271643, 425109, 660756, 1021170, 1568905, 2398059, 3646437, 5519025, 8314623, 12473538, 18634748, 27731820, 41113453, 60735830, 89411503, 131193675

Offset: 1

Wouter Meeussen, Jan 11 2004

nonn

Because of symmetry of the 3-dimensional-Ferrers plots, this sequence also counts the plane partitions with 3 or more rows and the plane partitions with maximal element >= 3

			a(4)=3:
  1111 111 211
  .... 1.. ...

Alois P. Heinz, Table of n, a(n) for n = 1..5000
Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000219, A000991, A091357.

b:= proc(n, k) option remember; `if`(n=0, 1, add(add(min(d, k)
      *d, d=numtheory[divisors](j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, infinity)-b(n,2):
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 24 2018

(* planepartitions[] : see link *); Table[Count[planepartitions[n], q_ /; Length[First[q]] >= 3], {n, 12}]

a(21)-a(27) from Vaclav Kotesovec, May 05 2018

a(28)-a(38) from Alois P. Heinz, Sep 24 2018

A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 7, 9, 5, 2, 1, 11, 18, 11, 5, 2, 1, 15, 30, 22, 11, 5, 2, 1, 22, 53, 42, 24, 11, 5, 2, 1, 30, 85, 78, 46, 24, 11, 5, 2, 1, 42, 139, 138, 90, 48, 24, 11, 5, 2, 1, 56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1, 77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1, 101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1

Offset: 1

Christian G. Bower, Jan 02 2004

Row sums give A000219.
Columns 1-5 are respectively A000041, A091356, A091357, A091358, and A091359.
Columns converge to A091360.

			Triangle starts:
01:  1,
02:  2, 1,
03:  3, 2, 1,
04:  5, 5, 2, 1,
05:  7, 9, 5, 2, 1,
06:  11, 18, 11, 5, 2, 1,
07:  15, 30, 22, 11, 5, 2, 1,
08:  22, 53, 42, 24, 11, 5, 2, 1,
09:  30, 85, 78, 46, 24, 11, 5, 2, 1,
10:  42, 139, 138, 90, 48, 24, 11, 5, 2, 1,
11:  56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1,
12:  77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1,
13:  101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1,
14:  135, 760, 1088, 899, 580, 336, 182, 96, 48, 24, 11, 5, 2, 1,
15:  176, 1115, 1741, 1512, 1020, 606, 340, 182, 96, 48, 24, 11, 5, 2, 1,
...

Alois P. Heinz, Rows n = 1..141, flattened

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      min(d, k)*d, d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Mar 15 2014

(* load EulerTransform from 'seqtranslib.m' under OEIS-Transforms *) Table[EulerTransform[Table[Min[c, r], {r, 20}]] - EulerTransform[Table[Min[c-1, r], {r, 20}]], {c, 20}] // Transpose
(* second program: *)
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k]*d, {d, Divisors[j]}] *A[n-j, k], {j, 1, n}]/n]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1] ]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)

k-th column is EulerTransform[1, 2, 3, .., k, k, k, ..]-EulerTransform[1, 2, 3, .., k-1, k-1, k-1, ..]. - Wouter Meeussen, Aug 29 2004

Definition corrected, Joerg Arndt, Jul 21 2014

A091356 Number of planar partitions of n with exactly 2 rows.

Original entry on oeis.org

1, 2, 5, 9, 18, 30, 53, 85, 139, 215, 336, 504, 760, 1115, 1635, 2351, 3375, 4770, 6725, 9368, 13006, 17885, 24510, 33319, 45139, 60743, 81457, 108610, 144334, 190844, 251542, 330082, 431825, 562710, 731154, 946644, 1222305, 1573155, 2019471

Offset: 2

Christian G. Bower, Jan 02 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Column 2 of A091355.

Cf. A000219, A091357, A091358, A091359, A091360.

b:= proc(n, k) option remember; `if`(n=0, 1, add(add(min(d, k)
     *d, d=numtheory[divisors](j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, 2)-b(n, 1):
seq(a(n), n=2..50);  # Alois P. Heinz, Oct 02 2018

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k] d, {d, Divisors[j]}] b[n - j, k], {j, 1, n}]/n];
a[n_] :=  b[n, 2] - b[n, 1];
a /@ Range[2, 50] (* Jean-François Alcover, Oct 28 2020, after Alois P. Heinz *)

a(n) = A000990(n) - A000041(n).

cp's OEIS Frontend

A089924 Number of plane partitions of n with 3 or more columns.

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A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.

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A091356 Number of planar partitions of n with exactly 2 rows.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula