A327117 -id:A327117 - cp's OEIS frontend

A116540 Number of zero-one matrices with n ones and no zero rows or columns, up to permutation of rows.

1, 1, 3, 10, 41, 192, 1025, 6087, 39754, 282241, 2159916, 17691161, 154192692, 1423127819, 13851559475, 141670442163, 1517880400352, 16989834719706, 198191448685735, 2404300796114642, 30273340418567819, 394948562421362392, 5330161943597341380, 74307324695105372519

Offset: 0

Vladeta Jovovic, Mar 27 2006

nonn

Also number of normal set multipartitions of weight n. These are defined as multisets of sets that together partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. Set multipartitions are involved in the expansion of elementary symmetric functions in terms of augmented monomial symmetric functions. - Gus Wiseman, Oct 22 2015

			The a(3) = 10 normal set multipartitions are: {1,1,1}, {1,12}, {1,1,2}, {2,12}, {1,2,2}, {123}, {1,23}, {2,13}, {3,12}, {1,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..230
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.
Gus Wiseman, Four symmetric function identities

Cf. A049311, A101370.

Row sums of A327117.

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)*binomial(binomial(k, i)+j-1, 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..24);  # Alois P. Heinz, Sep 13 2019

MSOSA[s_List] :=
  MSOSA[s] = If[Length[s] === 0, {{}}, Module[{sbs, fms},
     sbs = Rest[Subsets[Union[s]]];
     fms =
      Function[r,
        Append[#, r] & /@
         MSOSA[Fold[DeleteCases[#1, #2, {1}, 1] &, s, r]]] /@ sbs;
     Select[Join @@ fms, OrderedQ]
     ]];
mmallnorm[n_Integer] :=
  Function[s, Array[Count[s, y_ /; y <= #] + 1 &, n]] /@
   Subsets[Range[n - 1] + 1];
Array[Plus @@ Length /@ MSOSA /@ mmallnorm[#] &, 9]
(* Gus Wiseman, Oct 22 2015 *)

R(n, k)={Vec(-1 + 1/prod(j=1, k, (1 - x^j + O(x*x^n))^binomial(k, j) ))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

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).

A326914 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), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

Original entry on oeis.org

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, 3, 1596, 44880, 402756, 1668338, 3652712

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, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

Main diagonal gives A000110.
Row sums give A116539.
Column sums give A003465.
Cf. A001787, A255903, A326962 (this triangle read by columns), A327115, A327116, A327117.

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, 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), 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_] := 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], {k, h[n], n}], {n, 0, 12}] // 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.

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.

A327118 Total number of colors in all colored integer partitions of n using all colors of an initial color palette such that a color pattern for part i has i distinct colors in increasing order.

Original entry on oeis.org

0, 1, 5, 24, 129, 752, 4796, 33117, 246336, 1961233, 16626100, 149376533, 1416602126, 14130107135, 147781380186, 1616110614723, 18434515499407, 218849548323400, 2698686223271769, 34504328470389166, 456669361749612835, 6247290917385938422, 88216873775207493056

Offset: 0

Alois P. Heinz, Sep 13 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..230
Wikipedia, Partition (number theory)

Cf. 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-1, j), j=0..n/i)))
    end:
a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*C(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, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k, i]+j-1, j], {j, 0, n/i}]]];
a[n_] := Sum[k Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, May 06 2020, after Maple *)

a(n) = Sum_{k=1..n} k * A327117(n,k).

A327842 Number of colored integer partitions of n using all colors of a 3-set such that a color pattern for part i has i distinct colors in increasing order.

Original entry on oeis.org

5, 18, 45, 94, 174, 300, 486, 756, 1131, 1647, 2334, 3243, 4416, 5922, 7818, 10195, 13128, 16734, 21110, 26403, 32736, 40291, 49221, 59748, 72060, 86424, 103068, 122310, 144423, 169782, 198711, 231648, 268974, 311197, 358761, 412251, 472181, 539220, 613959

Offset: 3

Alois P. Heinz, Sep 27 2019

Alois P. Heinz, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Column k=3 of A327117.

G.f.: -(x^9-6*x^7+x^6+9*x^5+3*x^4-6*x^3-9*x^2+3*x+5)*x^3 / ((x^2+x+1) *(x+1)^3 *(x-1)^7).

A327843 Number of colored integer partitions of 2n using all colors of an n-set such that a color pattern for part i has i distinct colors in increasing order.

Original entry on oeis.org

1, 1, 7, 94, 2081, 67390, 2969647, 169299808, 12032189630, 1036485156029, 105880393642170, 12604896326749405, 1724189631362670619, 267831346979691504798, 46782781937811822181581, 9111872329195713764645644, 1964607669245374038857479576

Offset: 0

Alois P. Heinz, Sep 27 2019

nonn

			a(2) = 7: 2ab2ab, 2ab1a1a, 2ab1a1b, 2ab1b1b 1a1a1a1b, 1a1a1b1b, 1a1b1b1b.

Alois P. Heinz, Table of n, a(n) for n = 0..160

Cf. A327117.

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)*binomial(
      binomial(k, i)+j-1, j), j=0..n/i)))
    end:
a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..17);

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}]]];
a[n_] := Sum[b[2n, 2n, i] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

a(n) = A327117(2n,n).

cp's OEIS Frontend

A116540 Number of zero-one matrices with n ones and no zero rows or columns, 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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A326914 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), n>=0, min(j:A001787(j)>=n)<=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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A327118 Total number of colors in all colored integer partitions of n using all colors of an initial color palette such that a color pattern for part i has i distinct colors in increasing order.

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Mathematica

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A327842 Number of colored integer partitions of n using all colors of a 3-set such that a color pattern for part i has i distinct colors in increasing order.

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A327843 Number of colored integer partitions of 2n using all colors of an n-set such that a color pattern for part i has i distinct colors in increasing order.

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