A321878 -id:A321878 - cp's OEIS frontend

A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 6, 8, 5, 0, 1, 5, 8, 15, 14, 7, 0, 1, 6, 10, 24, 27, 24, 11, 0, 1, 7, 12, 35, 44, 51, 40, 15, 0, 1, 8, 14, 48, 65, 88, 93, 64, 22, 0, 1, 9, 16, 63, 90, 135, 176, 159, 100, 30, 0, 1, 10, 18, 80, 119, 192, 295, 312, 264, 154, 42, 0

Offset: 0

Alois P. Heinz, Aug 27 2019

			A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b.
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,    1,    1,    1, ...
  0,  1,   2,   3,   4,   5,    6,    7,    8, ...
  0,  2,   4,   6,   8,  10,   12,   14,   16, ...
  0,  3,   8,  15,  24,  35,   48,   63,   80, ...
  0,  5,  14,  27,  44,  65,   90,  119,  152, ...
  0,  7,  24,  51,  88, 135,  192,  259,  336, ...
  0, 11,  40,  93, 176, 295,  456,  665,  928, ...
  0, 15,  64, 159, 312, 535,  840, 1239, 1744, ...
  0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Jessica Jay and Benjamin Lees, Combinatorial identities from an inhomogeneous Ising chain, arXiv:2401.16311 [math.PR], 2024.
Wikipedia, Partition (number theory)

Columns k=0-4 give: A000007, A000041, A015128, A264686, A266821.
Main diagonal gives A321880.
Cf, A000142, A003056, A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

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

A(n,k) = Sum_{i=0..floor((sqrt(1+8*k)-1)/2)} k!/(k-i)! * A321878(n,i).

A321880 Number of partitions of n into colored blocks of equal parts with colors from a set of size n.

Original entry on oeis.org

1, 1, 4, 15, 44, 135, 456, 1239, 3424, 8694, 27240, 65846, 171864, 406133, 960848, 2615460, 5998416, 14304089, 32273100, 72271516, 153768520, 385905072, 817485768, 1841794483, 3915726528, 8388036950, 17125197336, 35051814558, 78986793592, 160176485813

Offset: 0

Alois P. Heinz, Aug 27 2019

nonn

			a(3) = 15: 3a, 3b, 3c, 2a1a, 2a1b, 2a1c, 2b1a, 2b1b, 2b1c, 2c1a, 2c1b, 2c1c, 111a, 111b, 111c.

Alois P. Heinz, Table of n, a(n) for n = 0..1650

Main diagonal of A321884.

Cf. A000142, A003056, A321878, A325916.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, k*add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i) +b(n, i-1, k)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..31);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := b[n, n, n];
a /@ Range[0, 31] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) = [x^n] Product_{j=1..n} (1+(n-1)*x^j)/(1-x^j).
a(n) = A321884(n,n).
a(n) = Sum_{i=0..floor((sqrt(1+8*n)-1)/2)} n!/(n-i)! * A321878(n,i).
a(n) = n * A325916(n) for n > 0, a(n) = 1.

A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 87, 138, 211, 324, 486, 727, 1079, 1584, 2305, 3337, 4789, 6830, 9712, 13689, 19225, 26841, 37322, 51598, 71108, 97580, 133350, 181558, 246335, 332991, 448706, 602607, 806732, 1077333, 1433885, 1903682, 2520246, 3328549, 4383929

Offset: 0

Gus Wiseman, May 25 2018

nonn

			The a(6) = 21 unitary factorizations:
(13) (21) (22) (25) (27) (28) (30) (36) (40) (48) (64)
(2*11) (2*15) (3*7) (3*10) (3*16) (4*7) (4*9) (5*6) (5*8)
(2*3*5)
The a(6) = 21 multiset partitions:
{{6}}
{{2,4}}
{{1,5}}
{{3,3}}
{{2,2,2}}
{{1,1,4}}
{{1,2,3}}
{{1,1,2,2}}
{{1,1,1,3}}
{{1,1,1,1,2}}
{{1,1,1,1,1,1}}
{{1},{5}}
{{1},{2,3}}
{{2},{4}}
{{2},{1,3}}
{{2},{1,1,1,1}}
{{1,1},{4}}
{{1,1},{2,2}}
{{3},{1,2}}
{{3},{1,1,1}}
{{1},{2},{3}}

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000110, A001055, A001221, A001970, A034444, A089233, A258466, A259936, A281116, A285572, A305078, A305079.

Row sums of A321878.

Table[Sum[BellB[Length[Union[y]]],{y,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz in A321878 *)

A322304 Total number of colors in all partitions of n into colored blocks of equal parts, such that all colors from a given set are used and the colors are introduced in increasing order.

Original entry on oeis.org

0, 1, 2, 5, 9, 17, 32, 55, 93, 154, 257, 407, 648, 1003, 1546, 2367, 3566, 5323, 7889, 11579, 16854, 24495, 35171, 50345, 71520, 101184, 142118, 198981, 277260, 384457, 530875, 730220, 1000192, 1365105, 1856155, 2514737, 3398397, 4574460, 6141309, 8218229

Offset: 0

Alois P. Heinz, Aug 28 2019

nonn

			a(4) = 9. The colored partitions are: 1111a, 2a11a, 22a, 3a1a, 4a, 2a11b, 3a1b.  The total number of colors used is 1+1+1+1+1+2+2 = 9.

Alois P. Heinz, Table of n, a(n) for n = 0..1650

Cf. A003056, A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= proc(n) option remember; add(add(binomial(k, i)*(-1)^i*
      b(n$2, k-i), i=0..k)/(k-1)!, k=1..floor((sqrt(1+8*n)-1)/2))
    end:
seq(a(n), n=0..44);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := Sum[Sum[Binomial[k, i] (-1)^i b[n, n, k - i], {i, 0, k}]/(k - 1)!, {k, 1, Floor[(Sqrt[1 + 8n] - 1)/2]}];
a /@ Range[0, 44] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..A003056(n)} k * A321878(n,k).

A327285 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size two are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 9, 17, 28, 47, 74, 116, 175, 263, 385, 560, 800, 1135, 1589, 2210, 3041, 4160, 5642, 7609, 10189, 13575, 17976, 23694, 31066, 40559, 52708, 68230, 87957, 112985, 144594, 184437, 234466, 297159, 375453, 473039, 594298, 744681, 930674, 1160271, 1442989

Offset: 3

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..5000

Column k=2 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(2):
seq(a(n), n=3..44);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := With[{k = 2}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[3, 44] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) ~ exp(Pi*sqrt(n)) / (16*n). - Vaclav Kotesovec, Sep 18 2019

A327286 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size three are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 10, 21, 37, 67, 112, 187, 302, 479, 741, 1136, 1707, 2539, 3732, 5424, 7804, 11133, 15743, 22088, 30774, 42582, 58540, 80007, 108725, 146955, 197646, 264525, 352433, 467541, 617651, 812734, 1065417, 1391592, 1811296, 2349775, 3038515, 3917052, 5034647

Offset: 6

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..5000

Column k=3 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(3):
seq(a(n), n=6..47);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := With[{k = 3}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[6, 47] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-2))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-2)) / (72*Pi*n). - Vaclav Kotesovec, Sep 18 2019

A327287 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size four are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 10, 20, 40, 72, 127, 217, 362, 587, 954, 1494, 2330, 3562, 5403, 8060, 11954, 17531, 25490, 36733, 52570, 74620, 105273, 147479, 205390, 284516, 391819, 536891, 732028, 993540, 1342174, 1805795, 2419115, 3228530, 4292484, 5686507, 7506642, 9877321

Offset: 10

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..5000

Column k=4 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(4):
seq(a(n), n=10..49);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := With[{k = 4}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[10, 49] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-3))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-3)) / (4*4!*sqrt(12)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

A327288 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size five are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 73, 125, 222, 372, 623, 1002, 1611, 2559, 3984, 6139, 9355, 14096, 21028, 31093, 45523, 66403, 95779, 137495, 195813, 277531, 390428, 546942, 761113, 1054749, 1454412, 1996271, 2727247, 3711683, 5029288, 6789347, 9130315, 12234596, 16335987

Offset: 15

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 15..5000

Column k=5 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(5):
seq(a(n), n=15..53);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[15, 53] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-4))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-4)) / (4*5!*sqrt(15)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

A327289 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size six are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 123, 210, 362, 603, 994, 1595, 2541, 3956, 6225, 9501, 14516, 21820, 32703, 48315, 71175, 103589, 150167, 216413, 309627, 440400, 623404, 877303, 1228493, 1712235, 2374639, 3278894, 4507571, 6175713, 8421243, 11447049, 15496728

Offset: 21

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 21..5000

Column k=6 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(6):
seq(a(n), n=21..59);

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-5))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-5)) / (4*6!*sqrt(18)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

A327290 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size seven are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 204, 337, 573, 934, 1527, 2416, 3826, 5907, 9088, 13963, 21070, 31642, 47131, 69707, 102214, 149143, 215754, 310547, 443840, 633139, 895294, 1262971, 1770236, 2473601, 3436809, 4761393, 6561269, 9015761, 12330231, 16812326

Offset: 28

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 28..5000

Column k=7 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(7):
seq(a(n), n=28..65);

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-6))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-6)) / (4*7!*sqrt(21)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

cp's OEIS Frontend

A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A321880 Number of partitions of n into colored blocks of equal parts with colors from a set of size n.

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A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

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Mathematica

A322304 Total number of colors in all partitions of n into colored blocks of equal parts, such that all colors from a given set are used and the colors are introduced in increasing order.

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A327285 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size two are used and the colors are introduced in increasing order.

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A327286 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size three are used and the colors are introduced in increasing order.

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A327287 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size four are used and the colors are introduced in increasing order.

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A327288 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size five are used and the colors are introduced in increasing order.

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