A319731 -id:A319731 - cp's OEIS frontend

A258466 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.

1, 1, 3, 8, 25, 82, 307, 1256, 5688, 28044, 149598, 855811, 5217604, 33711592, 229798958, 1646312694, 12355368849, 96861178984, 791258781708, 6720627124140, 59234364096426, 540812222095821, 5106663817156741, 49798678280227488, 500857393908312587

Offset: 0

Alois P. Heinz, May 30 2015

nonn

Also number of ways of partitioning a multiset with multiplicities some partition of n into disjoint blocks. Example: a(4) = 25: 1111; 111,2; 1112; 11,22; 1122; 11,2,3; 11,23; 112,3; 113,2; 1123; 1,2,3,4; 1,2,34; 1,23,4; 1,24,3; 1,234; 12,3,4; 12,34; 13,2,4; 13,24; 14,2,3; 14,23; 123,4; 124,3; 134,2; 1234. Formula: a(n) is the sum of Bell numbers of lengths of all integer partitions of n. - Gus Wiseman, Feb 17 2016

			a(3) = 8: 1a1a1a, 2a1a, 3a, 1a1a1b, 1a1b1a, 1a1b1b, 2a1b, 1a1b1c (in this example the sorts are labeled a, b, c).

Alois P. Heinz, Table of n, a(n) for n = 0..400

Row sums of A256130.

Cf. A000110, A035310, A262496, A278644, A319731.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..25);

Table[Plus @@ BellB /@ Length /@ IntegerPartitions[n], {n, 0, 24}] (* Gus Wiseman, Feb 17 2016 *)
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 01 2016, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} A256130(n,k).
a(n) ~ Bell(n) = A000110(n). - Vaclav Kotesovec, Jun 01 2015
G.f.: Sum_{k>=0} Bell(k) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020

A319730 Number T(n,k) of plane partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 0, 6, 11, 3, 0, 13, 48, 33, 5, 0, 24, 165, 212, 75, 7, 0, 48, 573, 1253, 798, 172, 11, 0, 86, 1759, 6114, 6175, 2284, 326, 15, 0, 160, 5473, 29573, 45040, 25697, 6198, 631, 22, 0, 282, 16051, 131488, 289685, 238516, 86189, 14519, 1102, 30

Offset: 0

Alois P. Heinz, Sep 26 2018

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,     2;
  0,   6,    11,      3;
  0,  13,    48,     33,      5;
  0,  24,   165,    212,     75,      7;
  0,  48,   573,   1253,    798,    172,    11;
  0,  86,  1759,   6114,   6175,   2284,   326,    15;
  0, 160,  5473,  29573,  45040,  25697,  6198,   631,   22;
  0, 282, 16051, 131488, 289685, 238516, 86189, 14519, 1102, 30;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Eric Weisstein's World of Mathematics, Plane partition
Wikipedia, Plane partition

Columns k=0-1 give: A000007, A000219 (for n>0).
Main diagonal gives A000041.
Row sums give A319731.
T(2n,n) gives A319732.
Cf. A256130, A319600.

T(n,k) = 1/k! * A319600(n,k).

cp's OEIS Frontend

A258466 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.

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