A326914 -id:A326914 - cp's OEIS frontend

A003465 Number of ways to cover an n-set.

1, 1, 5, 109, 32297, 2147321017, 9223372023970362989, 170141183460469231667123699502996689125, 57896044618658097711785492504343953925273862865136528166133547991141168899281

Offset: 0

N. J. A. Sloane

Let S be an n-element set, and let P be the set of all nonempty subsets of S. Then a(n) = number of subsets of P whose union is S.
Including the empty set doubles the entries, and we get A000371.
For disjoint covers see A000110. - Manfred Boergens, May 13 2024
For disjoint covers which may include one empty set see A186021. - Manfred Boergens, Mar 09 2025

			Let n=2, S={a,b}, P={a,b,ab}. There are five subsets of P whose union is S: {ab}, {a,b}, {a,ab}, {b,ab}, {a,b,ab}. - _Marc LeBrun_, Nov 10 2010

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 165.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. G. Wagner, Covers of finite sets, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 515-520.

Alois P. Heinz, Table of n, a(n) for n = 0..11
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
Martin Klazar, Extremal problems for ordered hypergraphs, arXiv:math/0305048 [math.CO], 2003.
Liwen Ma, Classification of coverings in the finite approximation spaces, Inf. Sci. 276 (2014) 31-41
A. J. Macula, Covers of a finite set, Math. Mag., 67 (1994), 141-144.
S. Spasovski and A. M. Bogdanova, Optimization of the Polynomial Greedy Solution for the Set Covering Problem, 2013, 10th Conference for Informatics and Information Technology (CIIT 2013).
Eric Weisstein's World of Mathematics, Cover.
Yoad Winter and Remko Scha, Plurals, draft chapter for the Wiley-Blackwell Handbook of Contemporary Semantics - second edition, edited by Shalom Lappin and Chris Fox, 2014.
Ping Zhou, Covering rough sets based on neighborhoods: an approach without using neighborhoods, Int. J. Approx. Reas. 52 (2011) 461-472, Section 3

Cf. A007537, A000371, A055154 (row sums), A369950 (diagonal for n>=1), A055621 (unlabeled case).
Column sums of A326914 and of A326962.
Cf. A000110, A186021.

a:= n-> add((-1)^k * binomial(n, k)*2^(2^(n-k))/2, k=0..n):
seq(a(n), n=0..11);  # Alois P. Heinz, Aug 24 2014

Table[Sum[(-1)^j Binomial[n,j] 2^(2^(n-j)-1),{j,0,n}],{n,0,10}] (* Geoffrey Critzer, Jun 26 2013 *)

{a(n) = sum(k=0, n, (-1)^k * n!/k!/(n-k)! * 2^(2^(n-k))) / 2} /* Michael Somos, Jun 14 1999 */

a(n) = Sum_{k>=0} (-1)^k * binomial(n, k) * 2^(2^(n-k)) / 2. - Michael Somos, Jun 14 1999
E.g.f.: (1/2)*Sum_{n>=0} exp((2^n-1)*x)*log(2)^n/n!. - Vladeta Jovovic, May 30 2004
a(n) ~ 2^(2^n - 1). - Vaclav Kotesovec, Jul 02 2016

More terms and comments from Michael Somos

Entry revised by N. J. A. Sloane, Nov 23 2010

A116539 Number of zero-one matrices with n ones and no zero rows or columns and with distinct rows, up to permutation of rows.

Original entry on oeis.org

1, 1, 2, 7, 28, 134, 729, 4408, 29256, 210710, 1633107, 13528646, 119117240, 1109528752, 10889570768, 112226155225, 1210829041710, 13640416024410, 160069458445202, 1952602490538038, 24712910192430620, 323964329622503527, 4391974577299578248, 61488854148194151940

Offset: 0

Vladeta Jovovic, Mar 27 2006

nonn

Also the number of labeled hypergraphs spanning an initial interval of positive integers with edge-sizes summing to n. - Gus Wiseman, Dec 18 2018

			From _Gus Wiseman_, Dec 18 2018: (Start)
The a(3) = 7 edge-sets:
    {{1,2,3}}
   {{1},{1,2}}
   {{2},{1,2}}
   {{1},{2,3}}
   {{2},{1,3}}
   {{3},{1,2}}
  {{1},{2},{3}}
Inequivalent representatives of the a(4) = 28 0-1 matrices:
  [1111]
.
  [100][1000][010][0100][001][0010][0001][110][110][1100][101][1010][1001]
  [111][0111][111][1011][111][1101][1110][101][011][0011][011][0101][0110]
.
  [10][100][100][1000][100][100][1000][1000][010][010][0100][0100][0010]
  [01][010][010][0100][001][001][0010][0001][001][001][0010][0001][0001]
  [11][101][011][0011][110][011][0101][0110][110][101][1001][1010][1100]
.
  [1000]
  [0100]
  [0010]
  [0001]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
P. J. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes , arXiv:math/0510155 [math.CO], 2005-2006.
M. Klazar, Extremal problems for ordered hypergraphs, arXiv:math/0305048 [math.CO], 2003.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763
Cf. A049311, A058891, A101370, A255906, A283877, A306021, A319190.
Row sums of A326914 and of A326962.

b:= proc(n, i, k) b(n, i, k):=`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j,
      min(n-i*j, i-1), k)*binomial(binomial(k, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 13 2019

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k]*Binomial[Binomial[k, i], j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

a(0)=1 prepended and more terms added by Alois P. Heinz, Sep 13 2019

A255903 Number T(n,k) of collections of nonempty multisets with a total of n objects of exactly k colors; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 8, 5, 0, 5, 23, 33, 15, 0, 7, 56, 141, 144, 52, 0, 11, 127, 492, 848, 675, 203, 0, 15, 268, 1518, 3936, 5190, 3396, 877, 0, 22, 547, 4320, 15800, 30710, 32835, 18270, 4140, 0, 30, 1072, 11567, 57420, 154410, 240012, 216006, 104656, 21147

Offset: 0

Alois P. Heinz, Mar 10 2015

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

In the case of exactly one color (k=1) each multiset of monochrome objects is fully described by its size and a collection of sizes corresponds to an integer partition. In the case of distinct colors for all objects (k=n) every multiset collection is a set partition.

			T(3,1) = 3: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}.
T(3,2) = 8: {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}.
T(3,3) = 5: {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,   8,    5;
  0,  5,  23,   33,    15;
  0,  7,  56,  141,   144,    52;
  0, 11, 127,  492,   848,   675,   203;
  0, 15, 268, 1518,  3936,  5190,  3396,   877;
  0, 22, 547, 4320, 15800, 30710, 32835, 18270, 4140;
  ...

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

Columns k=0-10 give: A000007, A000041 (for n>0), A255942, A255943, A255944, A255945, A255946, A255947, A255948, A255949, A255950.
Main and lower diagonals give: A000110, A255951, A255952, A255953, A255954, A255955, A255956, A255957, A255958, A255959, A255960.
Row sums give A255906.
Antidiagonal sums give A258450.
T(2n,n) gives A255907.
Cf. A075196, A317178, A326914, A326962, A327116, A327117.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      add(d*binomial(d+k-1, k-1), d=divisors(j)), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k]*Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i * Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12} ] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A075196(n,k-i).

Sum_{k=0..n} k * T(n,k) = A317178(n).

A327116 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 2, 15, 27, 15, 0, 3, 32, 102, 124, 52, 0, 4, 65, 319, 656, 600, 203, 0, 5, 124, 897, 2780, 4210, 3084, 877, 0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140, 0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147

Offset: 0

Alois P. Heinz, Sep 13 2019

			T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   2;
  0, 2,   6,    5;
  0, 2,  15,   27,    15;
  0, 3,  32,  102,   124,     52;
  0, 4,  65,  319,   656,    600,    203;
  0, 5, 124,  897,  2780,   4210,   3084,    877;
  0, 6, 230, 2346, 10305,  23040,  27567,  16849,  4140;
  0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition (number theory)

Columns k=0-2 give: A000007, A000009 (for n>0), A327598.
Main diagonal gives A000110.
Row sums give A317776.
T(2n,n) gives A327556.
Cf. A255903, A326914, A326962, A327117, A327557, A309973.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

c = Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327557(n).

A326962 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.

Original entry on oeis.org

1, 1, 2, 2, 1, 5, 12, 18, 20, 18, 15, 11, 6, 3, 1, 15, 64, 166, 332, 566, 864, 1214, 1596, 1975, 2320, 2600, 2780, 2842, 2780, 2600, 2320, 1979, 1608, 1238, 908, 626, 404, 246, 136, 69, 32, 12, 4, 1, 52, 340, 1315, 3895, 9770, 21848, 44880, 86275, 157140

Offset: 0

Alois P. Heinz, Sep 13 2019

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1, 12,   15;
           18,   64,    52;
           20,  166,   340,    203;
           18,  332,  1315,   1866,    877;
           15,  566,  3895,   9930,  10710,   4140;
           11,  864,  9770,  39960,  74438,  64520,  21147;
            6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;
  ...

Alois P. Heinz, Columns k = 0..10, flattened
Wikipedia, Partition (number theory)

Main diagonal gives A000110.
Row sums give A116539.
Column sums give A003465.
Cf. A001787, A255903, A326914 (this triangle read by rows), A327115, A327116, A327117.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);

c = Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {i, 0, k}];
Table[Table[T[n, k], {n, k, k 2^(k-1)}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327115(n).
T(n*2^(n-1),n) = T(A001787(n),n) = 1.
T(n*2^(n-1)-1,n) = n for n >= 2.

A327117 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 7, 18, 15, 0, 1, 10, 45, 84, 52, 0, 1, 14, 94, 298, 415, 203, 0, 1, 18, 174, 844, 1995, 2178, 877, 0, 1, 23, 300, 2081, 7440, 13638, 12131, 4140, 0, 1, 28, 486, 4652, 23670, 64898, 95823, 71536, 21147, 0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975

Offset: 0

Alois P. Heinz, Sep 13 2019

The sequence of column k satisfies a linear recurrence with constant coefficients of order k*2^(k-1) = A001787(k).

			T(3,2) = 4: 2ab1a, 2ab1b, 1a1a1b, 1a1b1b.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  4,   5;
  0, 1,  7,  18,   15;
  0, 1, 10,  45,   84,    52;
  0, 1, 14,  94,  298,   415,    203;
  0, 1, 18, 174,  844,  1995,   2178,    877;
  0, 1, 23, 300, 2081,  7440,  13638,  12131,   4140;
  0, 1, 28, 486, 4652, 23670,  64898,  95823,  71536,  21147;
  0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition (number theory)

Columns k=0-3 give: A000007, A057427, A014616(n-1) for n>1, A327842.
Main diagonal gives A000110.
Row sums give A116540.
T(2n,n) gives A327843.
Cf. A001787, A255903, A326914, A326962, A327116, A327118.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i)+j-1, j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] Binomial[Binomial[k, i] + j - 1, j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327118(n).

A327583 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order; triangle T(n,k), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 3, 4, 13, 6, 48, 75, 150, 536, 541, 300, 2820, 6320, 4683, 666, 11144, 50150, 81012, 47293, 936, 41346, 308080, 903210, 1134952, 545835, 1824, 131304, 1689040, 7805080, 16914786, 17346352, 7087261, 2520, 420084, 8279640, 58728660, 194051060, 333027016

Offset: 0

Alois P. Heinz, Sep 17 2019

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(3,2) = 4: 2ab1a, 2ab1b, 1a2ab, 1b2ab.
T(3,3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
T(4,2) = 6: 2ab1a1b, 1a2ab1b, 1a1b2ab, 2ab1b1a, 1b2ab1a, 1b1a2ab.
Triangle T(n,k) begins:
  1;
     1;
        3;
        4,  13;
        6,  48,    75;
           150,   536,    541;
           300,  2820,   6320,   4683;
           666, 11144,  50150,  81012,   47293;
           936, 41346, 308080, 903210, 1134952, 545835;
           ...

Alois P. Heinz, Rows n = 0..150, flattened

Main diagonal gives A000670.
Row sums give A321587.
Column sums give A327585.
Cf. A001787, A326914, A327584 (this triangle read by columns).

C:= binomial:
g:= proc(n) option remember; n*2^(n-1) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!,
      `if`(i<1 or k add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=h(n)..n), n=0..12);

c = Binomial;
g[n_] := g[n] = n*2^(n - 1);
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0,
     h[n - 1]], True, k++, If[g[k] >= n, Return[k]]]];
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!,
     If[i < 1 || k < h[n], 0, Sum[b[n - i*j, Min[n - i*j, i - 1],
     k, p + j]*c[c[k, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k - i)*c[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2021, after Alois P. Heinz *)

A327115 Total number of colors used in all colored integer partitions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order.

Original entry on oeis.org

0, 1, 4, 19, 98, 570, 3642, 25292, 189454, 1519648, 12978141, 117437020, 1121299471, 11256640012, 118443403699, 1302670531063, 14938986954323, 178248001223476, 2208487163394749, 28363722744050886, 376991516806826090, 5178009641895235269, 73396161423153313320

Offset: 0

Alois P. Heinz, Sep 13 2019

nonn

			a(2) = 4: 2ab, 1a1b.  Both colors (a and b) are used twice: 2 + 2 = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A326914, A326962.

C:= binomial:
g:= proc(n) option remember; n*2^(n-1) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=h(n)..n):
seq(a(n), n=0..23);

c = Binomial;
g[n_] := g[n] = n 2^(n - 1);
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]] , True, k++, If [g[k] >= n ,  Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < h[n], 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];
a[n_] := Sum[k Sum[b[n, n, i] (-1)^(k-i) c[k, i], {i, 0, k}], {k, h[n], n}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} A326914(n,k) = Sum_{k=1..n} A326962(n,k).

cp's OEIS Frontend

A003465 Number of ways to cover an n-set.

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A116539 Number of zero-one matrices with n ones and no zero rows or columns and with distinct rows, up to permutation of rows.

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A255903 Number T(n,k) of collections of nonempty multisets with a total of n objects of exactly k colors; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A327116 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A326962 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.

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A327117 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A327583 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order; triangle T(n,k), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

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