A255970 -id:A255970 - cp's OEIS frontend

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0

Offset: 0

Alois P. Heinz, Sep 08 2014

In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - Vaclav Kotesovec, Mar 19 2015

When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field of size k. - Geoffrey Critzer, Nov 11 2022

			A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6,      7, ...
  0,  2,   6,   12,    20,     30,     42,     56, ...
  0,  3,  14,   39,    84,    155,    258,    399, ...
  0,  5,  34,  129,   356,    805,   1590,   2849, ...
  0,  7,  74,  399,  1444,   4055,   9582,  19999, ...
  0, 11, 166, 1245,  5876,  20455,  57786, 140441, ...
  0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Columns k=0-10 give: A000007, A000041, A070933, A242587, A246936, A246937, A246938, A246939, A246940, A246941, A246942.
Rows n=0-4 give: A000012, A001477, A002378, A027444, A186636.
Main diagonal gives A124577.
Cf. A144064, A255970, A256130.

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:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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]]]]; A[n_, k_] := b[n, n, k];  Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)

G.f. of column k: Product_{i>=1} 1/(1-k*x^i).

T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i).

A256130 Number T(n,k) of 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, 2, 1, 0, 3, 4, 1, 0, 5, 12, 7, 1, 0, 7, 30, 33, 11, 1, 0, 11, 72, 130, 77, 16, 1, 0, 15, 160, 463, 438, 157, 22, 1, 0, 22, 351, 1557, 2216, 1223, 289, 29, 1, 0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1, 0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1

Offset: 0

Alois P. Heinz, Mar 15 2015

In general, column k>1 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - Vaclav Kotesovec, Jun 01 2015

			T(3,1) = 3: 1a1a1a, 2a1a, 3a.
T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b.
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,    1;
  0,  3,    4,     1;
  0,  5,   12,     7,     1;
  0,  7,   30,    33,    11,     1;
  0, 11,   72,   130,    77,    16,     1;
  0, 15,  160,   463,   438,   157,    22,    1;
  0, 22,  351,  1557,  2216,  1223,   289,   29,   1;
  0, 30,  743,  5031, 10422,  8331,  2957,  492,  37,  1;
  0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000041 (for n>0), A258457, A258458, A258459, A258460, A258461, A258462, A258463, A258464, A258465.
Row sums give A258466.
T(2n,n) give A258467.
Cf. A000142, A246935, A255970, A262495, A319730.

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):
seq(seq(T(n, k), k=0..n), n=0..10);

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}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *)

T(n,k) = A255970(n,k)/k! = (Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i)) / A000142(k).

A319600 Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 4, 0, 6, 22, 18, 0, 13, 96, 198, 120, 0, 24, 330, 1272, 1800, 840, 0, 48, 1146, 7518, 19152, 20640, 7920, 0, 86, 3518, 36684, 148200, 274080, 234720, 75600, 0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040, 0, 282, 32102, 788928, 6952440, 28621920, 62056080, 73175760, 44432640, 10886400

Offset: 0

Alois P. Heinz, Sep 24 2018

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,     4;
  0,   6,    22,     18;
  0,  13,    96,    198,     120;
  0,  24,   330,   1272,    1800,     840;
  0,  48,  1146,   7518,   19152,   20640,    7920;
  0,  86,  3518,  36684,  148200,  274080,  234720,   75600;
  0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040;
  ...

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).
Row sums give A319601.
Main diagonal gives A053529.
Cf. A255970, A319730.

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A306100(n,k-i).

T(n,k) = k! * A319730(n,k).

A278644 Number of partitions of n into parts of sorts {1, 2, ... }.

Original entry on oeis.org

1, 1, 4, 17, 95, 649, 5423, 53345, 604570, 7744990, 110596370, 1740967790, 29943077149, 558541778035, 11229820022013, 242071441524480, 5568954194762675, 136181762611151941, 3527284819779421843, 96465042641948254298, 2777679881076121497601

Offset: 0

Alois P. Heinz, Nov 24 2016

nonn

Parts are unordered, sorts are ordered, all sorts up to the highest have to be present.

a(n) mod 2 = A040051(n).

			a(3) = 17: 1a1a1a, 2a1a, 1a, 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a (in this example the sorts are labeled a, b, c).

Alois P. Heinz, Table of n, a(n) for n = 0..424

Row sums of A255970.

Cf. A000670, A008284, A040051, A258466.

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:
a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);

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]]]]; a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 06 2017, translated from Maple *)

a(n) = Sum_{k=0..n} A255970(n,k).
a(n) = Sum_{k=0..n} A008284(n,k) * A000670(k). - Ludovic Schwob, Sep 25 2023
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 26 2023

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A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A256130 Number T(n,k) of 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.

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A319600 Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A278644 Number of partitions of n into parts of sorts {1, 2, ... }.

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