A131632 -id:A131632 - cp's OEIS frontend

A007837 Number of partitions of n-set with distinct block sizes.

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847

Offset: 0

Arnold Knopfmacher

nonn

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..700
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Fig. 3, arXiv:math/0606370 [math.CO], 2006.
Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011

nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)

{my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Christian G. Bower

a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A226873 Number A(n,k) of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 4, 1, 0, 1, 1, 3, 10, 11, 1, 0, 1, 1, 3, 10, 23, 16, 1, 0, 1, 1, 3, 10, 47, 66, 42, 1, 0, 1, 1, 3, 10, 47, 126, 222, 64, 1, 0, 1, 1, 3, 10, 47, 246, 522, 561, 163, 1, 0, 1, 1, 3, 10, 47, 246, 882, 1821, 1647, 256, 1, 0

Offset: 0

Alois P. Heinz, Jun 21 2013

			A(4,3) = 23: aaaa, aaab, aaba, aabb, aabc, aacb, abaa, abab, abac, abba, abca, acab, acba, baaa, baab, baac, baba, baca, bbaa, bcaa, caab, caba, cbaa.
Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,    1,     1, ...
  0, 1,  1,   1,    1,    1,    1,     1, ...
  0, 1,  3,   3,    3,    3,    3,     3, ...
  0, 1,  4,  10,   10,   10,   10,    10, ...
  0, 1, 11,  23,   47,   47,   47,    47, ...
  0, 1, 16,  66,  126,  246,  246,   246, ...
  0, 1, 42, 222,  522,  882, 1602,  1602, ...
  0, 1, 64, 561, 1821, 3921, 6441, 11481, ...

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

Columns k=0-10 give: A000007, A000012, A027306, A092255, A092429, A226875, A226876, A226877, A226878, A226879, A226880.
Main diagonal gives: A005651.
Cf. A131632, A182172.

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

b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n-j, j, t-1]/j!, {j, i, n/t}]]; a[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A(n,k) = Sum_{i=0..min(n,k)} A226874(n,i).

A226874 Number T(n,k) of n-length words w over a k-ary alphabet {a1, a2, ..., ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w; 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, 3, 6, 0, 1, 10, 12, 24, 0, 1, 15, 50, 60, 120, 0, 1, 41, 180, 300, 360, 720, 0, 1, 63, 497, 1260, 2100, 2520, 5040, 0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320, 0, 1, 255, 5154, 20916, 58464, 90720, 151200, 181440, 362880

Offset: 0

Alois P. Heinz, Jun 21 2013

T(n,k) is the sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a multiset of size k.

			T(4,2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.
T(4,3) = 12: aabc, aacb, abac, abca, acab, acba, baac, baca, bcaa, caab, caba, cbaa.
T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   2;
  0,  1,   3,    6;
  0,  1,  10,   12,   24;
  0,  1,  15,   50,   60,   120;
  0,  1,  41,  180,  300,   360,   720;
  0,  1,  63,  497, 1260,  2100,  2520,  5040;
  0,  1, 162, 1484, 6496, 10080, 16800, 20160, 40320;
  ...

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

Columns k=0-10 give: A000007, A057427, A226881, A226882, A226883, A226884, A226885, A226886, A226887, A226888, A226889.
Main diagonal gives: A000142.
Row sums give: A005651.
T(2n,n) gives A318796.
Cf. A131632, A285824, A292222, A327803.

b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
T:= (n, k)-> `if`(n*k=0, `if`(n=k, 1, 0), n!*b(n, 1, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(x^j*b(n-i*j, i-1)*
      combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);

b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from first Maple *)

T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
{my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ Andrew Howroyd, Dec 20 2017

T(n,k) = A226873(n,k) - [k>0] * A226873(n,k-1).

A032011 Partition n labeled elements into sets of different sizes and order the sets.

Original entry on oeis.org

1, 1, 1, 7, 9, 31, 403, 757, 2873, 12607, 333051, 761377, 3699435, 16383121, 108710085, 4855474267, 13594184793, 76375572751, 388660153867, 2504206435681, 20148774553859, 1556349601444477, 5050276538344665, 33326552998257031, 186169293932977115, 1305062351972825281, 9600936552132048553, 106019265737746665727, 12708226588208611056333, 47376365554715905155127

Offset: 0

Christian G. Bower, Apr 01 1998

nonn

From Alois P. Heinz, Sep 02 2015: (Start)
Also the number of matrices with n rows of nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to n and the column sums are distinct.  Equivalently, the number of compositions of n into distinct parts where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once.
a(3) = 7:
[1]   [1 0]  [0 1]  [1 0]  [0 1]  [0 1]  [1 0]
[1]   [1 0]  [0 1]  [0 1]  [1 0]  [1 0]  [0 1]
[1]   [0 1]  [1 0]  [1 0]  [0 1]  [1 0]  [0 1].
3abc, 2ab1c, 1c2ab, 2ac1b, 1b2ac, 2bc1a, 1a2bc.  (End)

Alois P. Heinz, Table of n, a(n) for n = 0..670
C. G. Bower, Transforms (2)

Cf. A000670, A007837, A032020, A114902, A120774, A131632.

Main diagonal of A261836 and A261959.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 02 2015

f[list_]:=Apply[Multinomial,list]*Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 1, 30}]
b[n_, i_, p_] := b[n, i, p] = If[i*(i+1)/2n, 0, b[n-i, i-1, p+1]*Binomial[n, i]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

seq(n)=[subst(serlaplace(y^0*p),y,1) | p <- Vec(serlaplace(prod(k=1, n, 1 + x^k*y/k! + O(x*x^n))))] \\ Andrew Howroyd, Sep 13 2018

"AGJ" (ordered, elements, labeled) transform of 1, 1, 1, 1, ...

a(n) = Sum_{k>=0} k! * A131632(n,k). - Alois P. Heinz, Sep 09 2015

a(0)=1 prepended by Alois P. Heinz, Sep 02 2015

A327803 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 7, 3, 0, 31, 16, 0, 121, 125, 0, 831, 711, 60, 0, 5041, 5915, 525, 0, 42911, 46264, 6328, 0, 364561, 438681, 67788, 0, 3742453, 4371085, 753420, 12600, 0, 39916801, 49321745, 8924685, 166320, 0, 486891175, 588219523, 113501784, 2966040

Offset: 0

Alois P. Heinz, Sep 25 2019

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       3;
  0,       7,       3;
  0,      31,      16;
  0,     121,     125;
  0,     831,     711,     60;
  0,    5041,    5915,    525;
  0,   42911,   46264,   6328;
  0,  364561,  438681,  67788;
  0, 3742453, 4371085, 753420, 12600;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Columns k=0-2 give: A000007, A061095, A327826.
Row sums give A005651.
Cf. A000217, A003056, A022915, A131632 (when the parts are distinct), A226874.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> nops({x[]})=k, partition(n))):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
      combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[x^Sign[j]*b[n - i*j, i-1]*multinomial[n, Join[{n-i*j}, Table[i, {j}]]], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 06 2020, after 2nd Maple program *)

T(n*(n+1)/2,n) = T(A000217(n),n) = A022915(n).

A272514 Number of set partitions of [n] into two blocks with distinct sizes.

Original entry on oeis.org

3, 4, 15, 21, 63, 92, 255, 385, 1023, 1585, 4095, 6475, 16383, 26332, 65535, 106761, 262143, 431909, 1048575, 1744435, 4194303, 7036529, 16777215, 28354131, 67108863, 114159427, 268435455, 459312151, 1073741823, 1846943452, 4294967295, 7423131481, 17179869183

Offset: 3

Alois P. Heinz, May 01 2016

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=2 of A131632.

[(&+[Binomial(n,j): j in [1..Floor((n-1)/2)]]): n in [3..40]]; // G. C. Greubel, Jul 14 2024

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 2):
seq(a(n), n=3..40);

Table[Sum[Binomial[n, i], {i, Floor[(n - 1)/2]}], {n, 3, 35}] (* Michael De Vlieger, Nov 15 2017 *)

def A272514(n): return sum( binomial(n,j) for j in range(1,1+((n-1)//2)))
[A272514(n) for n in range(3,31)] # G. C. Greubel, Jul 14 2024

a(n) = n! * [x^n*y^2] Product_{n>=1} (1+y*x^n/n!).
a(n) = Sum_{i=1..floor((n-1)/2)} binomial(n,i). - Wesley Ivan Hurt, Nov 15 2017
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Dec 11 2020

A272515 Number of set partitions of [n] into three blocks with distinct sizes.

Original entry on oeis.org

60, 105, 448, 2016, 4980, 15675, 61644, 155155, 482573, 1733550, 4549808, 13890360, 48104628, 128949675, 392009140, 1322692581, 3607864403, 10929721440, 36245555284, 100109572875, 302709337515, 990788537700, 2763564406113, 8344789976616, 27039048750600

Offset: 6

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=3 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=6..40);

b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
a[n_] := b[n, n, 3];
a /@ Range[6, 40] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) = n! * [x^n*y^3] Product_{n>=1} (1+y*x^n/n!).

Conjecture: a(n) ~ 3^n / 6. - Vaclav Kotesovec, Dec 11 2020

A272516 Number of set partitions of [n] into four blocks with distinct sizes.

Original entry on oeis.org

12600, 27720, 138600, 643500, 4408404, 12687675, 60780720, 238299880, 1295666424, 4208874756, 18840460800, 72351683460, 361100656224, 1228553894491, 5370616442928, 20605640103400, 97659853077800, 342942099783075, 1479570975628200, 5678915129142255

Offset: 10

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=4 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 4):
seq(a(n), n=10..40);

b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
a[n_] := b[n, n, 4];
a /@ Range[10, 40] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) = n! * [x^n*y^4] Product_{n>=1} (1+y*x^n/n!).

A272517 Number of set partitions of [n] into five blocks with distinct sizes.

Original entry on oeis.org

37837800, 100900800, 588107520, 2977294320, 20020160160, 164118754800, 635661248040, 3295178686800, 17741374681800, 95826446465904, 623399389674600, 2664090278249400, 13876038856379700, 71797074694745400, 375274098870636420, 2199911433079733100

Offset: 15

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 15..1000

Column k=5 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=15..40);

b[n_, i_, t_] := b[n, i, t] = If[t > i || t(t+1)/2 > n || t(2i+1-t)/2 < n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t - 1]* Binomial[n, i]]]];
a[n_] := b[n, n, 5];
a /@ Range[15, 40] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) = n! * [x^n*y^5] Product_{n>=1} (1+y*x^n/n!).

A272518 Number of set partitions of [n] into six blocks with distinct sizes.

Original entry on oeis.org

2053230379200, 6453009763200, 43288940494800, 242418066770880, 1707999012720000, 12887361202716000, 144924867388501200, 620550897351184800, 4048435123506774000, 23424084614648718000, 161250104584826056800, 1013722794731975328000, 8616255173755280251200

Offset: 21

Alois P. Heinz, May 01 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 21..1000

Column k=6 of A131632.

b:= proc(n, i, t) option remember; `if`(t>i or t*(t+1)/2>n
      or t*(2*i+1-t)/2n, 0, b(n-i, i-1, t-1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 6):
seq(a(n), n=21..40);

a(n) = n! * [x^n*y^6] Product_{n>=1} (1+y*x^n/n!).

cp's OEIS Frontend

A007837 Number of partitions of n-set with distinct block sizes.

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A226873 Number A(n,k) of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

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A226874 Number T(n,k) of n-length words w over a k-ary alphabet {a1, a2, ..., ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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Maple

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PARI

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A032011 Partition n labeled elements into sets of different sizes and order the sets.

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PARI

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A327803 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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A272514 Number of set partitions of [n] into two blocks with distinct sizes.

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Magma

Maple

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SageMath

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A272515 Number of set partitions of [n] into three blocks with distinct sizes.

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