A327869 -id:A327869 - cp's OEIS frontend

A327870 Row sums of A327869.

1, 2, 2, 11, 14, 47, 305, 611, 2070, 8831, 84077, 204371, 944333, 3850407, 23991739, 297448526, 927586630, 4775902567, 24534836837, 141681919871, 1080484165089, 18553632475991, 66762080435239, 415657332495526, 2298883231736949, 15799818116227747

Offset: 0

Alois P. Heinz, Sep 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Row sums of A327869.

Cf. A320566.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$2, 0)*(n+1) -add(b(n$2, k), k=1..n):
seq(a(n), n=0..28);

b[n_, i_, k_] := b[n, i, k] = If[i*(i + 1)/2 < n, 0,
     If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] +
     If[i == k, 0, b[n - i, Min[n - i, i - 1], k]*Binomial[n, i]]]];
a[n_] := b[n, n, 0]*(n + 1) - Sum[b[n, n, k], {k, 1, n}];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Mar 28 2022, after Alois P. Heinz *)

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

Original entry on oeis.org

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

A327801 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 47, 40, 18, 4, 1, 246, 235, 100, 30, 5, 1, 1602, 1476, 705, 200, 45, 6, 1, 11481, 11214, 5166, 1645, 350, 63, 7, 1, 95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1, 871030, 859527, 413316, 134568, 30996, 5922, 840, 108, 9, 1

Offset: 0

Alois P. Heinz, Sep 25 2019

Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.

			Triangle T(n,k) begins:
      1;
      1,     1;
      3,     2,     1;
     10,     9,     3,     1;
     47,    40,    18,     4,    1;
    246,   235,   100,    30,    5,   1;
   1602,  1476,   705,   200,   45,   6,  1;
  11481, 11214,  5166,  1645,  350,  63,  7, 1;
  95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1;
  ...

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

Columns k=0-2 give: A005651, A327827, A327828.
Row sums give A320566.
T(2n,n) gives A266518.
T(n,n-1) gives A001477.
T(n+1,n-1) gives A045943.
Cf. A327869.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> k=0 or k in x, partition(n))):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
      `if`(i=k, 0, b(n-i, min(n-i, i), k)/i!))
    end:
T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, 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 < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i], k]/i!]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, from 2nd Maple program *)

A327884 Number T(n,k) of set partitions of [n] such that at least one of the block sizes is k or k=0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 4, 3, 1, 15, 11, 9, 4, 1, 52, 41, 35, 20, 5, 1, 203, 162, 150, 90, 30, 6, 1, 877, 715, 672, 455, 175, 42, 7, 1, 4140, 3425, 3269, 2352, 1015, 280, 56, 8, 1, 21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1, 115975, 98253, 97155, 76540, 39480, 12978, 3150, 600, 90, 10, 1

Offset: 0

Alois P. Heinz, Sep 28 2019

			T(4,1) = 11: 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,3) = 4: 123|4, 124|3, 134|2, 1|234.
T(4,4) = 1: 1234.
T(5,1) = 41: 1234|5, 1235|4, 123|4|5, 1245|3, 124|3|5, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 12|3|4|5, 1345|2, 134|2|5, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
      1;
      1,     1;
      2,     1,     1;
      5,     4,     3,     1;
     15,    11,     9,     4,    1;
     52,    41,    35,    20,    5,    1;
    203,   162,   150,    90,   30,    6,   1;
    877,   715,   672,   455,  175,   42,   7,  1;
   4140,  3425,  3269,  2352, 1015,  280,  56,  8, 1;
  21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Iverson bracket
Wikipedia, Partition of a set

Columns k=0-3 give: A000110, A000296(n+1), A327885, A328153.
T(2n,n) gives A276961.
Cf. A080510, A327869.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
    end:
T:= (n, k)-> b(n, 0)-`if`(k=0, 0, b(n, k)):
seq(seq(T(n, k), k=0..n), n=0..11);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[If[j == k, 0, b[n - j,k] Binomial[ n - 1, j - 1]], {j, 1, n}]];
T[n_, k_] := b[n, 0] - If[k == 0, 0, b[n, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

E.g.f. of column k: exp(exp(x)-1) - [k>0] * exp(exp(x)-1-x^k/k!).

T(n,0) - T(n,1) = A000296(n).

A328156 Number of set partitions of [2n] with distinct block sizes and one of the block sizes is n.

Original entry on oeis.org

1, 0, 0, 60, 280, 3780, 74844, 576576, 6949800, 110416020, 3319141540, 31333878576, 545777101324, 8349081650000, 196469122903200, 8108831645948160, 99934219113287400, 1961077012271694900, 39215221761564594900, 860948656518718429200, 25274389422461123124180

Offset: 0

Alois P. Heinz, Oct 05 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..510
Wikipedia, Partition of a set

Cf. A266518, A276961, A327869.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 (2*n)!*(b(2*n$2, 0)-`if`(n=0, 0, b(2*n$2, n))):
seq(a(n), n=0..22);

b[n_, i_, k_] := b[n, i, k] = If[i (i + 1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k]/i!]]];
a[n_] := (2n)! (b[2n, 2n, 0] - If[n == 0, 0, b[2n, 2n, n]]);
a /@ Range[0, 22] (* Jean-François Alcover, May 02 2020, after Maple *)

a(n) = A327869(2n,n).

A328155 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 3.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 60, 35, 336, 1848, 16080, 33825, 93280, 539396, 3856216, 49390250, 147478800, 708041160, 2354289744, 18196716309, 150847235040, 2615953578700, 9488756856040, 57565330671310, 296745669669768, 1435526275752900, 12231628020365000

Offset: 0

Alois P. Heinz, Oct 05 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..697
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
Wikipedia, Partition of a set

Column k=3 of A327869.

Cf. A328153.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$2, 0)-b(n$2, 3):
seq(a(n), n=0..29);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k]*Binomial[n, i]]]];
a[n_] := b[n, n, 0] - b[n, n, 3];
a /@ Range[0, 29] (* Jean-François Alcover, May 04 2020, after Maple *)

A327876 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 1.

Original entry on oeis.org

0, 1, 0, 3, 4, 5, 66, 112, 456, 765, 15070, 31856, 150756, 663962, 1943046, 44316105, 127348864, 661449549, 3220447446, 20913769072, 68889553260, 2403704171190, 7894983374674, 51012052828947, 270186003789312, 1836634462055350, 13402212376611266

Offset: 0

Alois P. Heinz, Sep 28 2019

nonn

Sum of multinomials M(n; lambda), where lambda ranges over all integer partitions of n into distinct parts and one part is 1.

Alois P. Heinz, Table of n, a(n) for n = 0..697
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
Wikipedia, Partition of a set

Column k=1 of A327869.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$2, 0)-b(n$2, 1):
seq(a(n), n=0..29);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k] Binomial[n, i]]]];
a[n_] := b[n, n, 0] - b[n, n, 1];
a /@ Range[0, 29] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

a(1) = 1: 1.
a(2) = 0.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 4: 123|4, 124|3, 134|2, 1|234.
a(5) = 5: 1234|5, 1235|4, 1245|3, 1345|2, 1|2345.
a(6) = 66: 12345|6, 12346|5, 12356|4, 123|45|6, 123|46|5, 123|4|56, 12456|3, 124|35|6, 124|36|5, 124|3|56, 125|34|6, 12|345|6, 126|34|5, 12|346|5, 125|36|4, 125|3|46, 126|35|4, 12|356|4, 126|3|45, 12|3|456, 13456|2, 134|25|6, 134|26|5, 134|2|56, 135|24|6, 13|245|6, 136|24|5, 13|246|5, 135|26|4, 135|2|46, 136|25|4, 13|256|4, 136|2|45, 13|2|456, 145|23|6, 14|235|6, 146|23|5, 14|236|5, 15|234|6, 1|23456, 16|234|5, 1|234|56, 156|23|4, 15|236|4, 16|235|4, 1|235|46, 1|236|45, 1|23|456, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 146|2|35, 14|2|356, 156|24|3, 15|246|3, 16|245|3, 1|245|36, 1|246|35, 1|24|356, 156|2|34, 15|2|346, 1|256|34, 1|25|346, 16|2|345, 1|26|345.

A327881 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 2.

Original entry on oeis.org

0, 0, 1, 3, 0, 10, 75, 126, 196, 1548, 15525, 39820, 161106, 358722, 3705884, 46623045, 142988280, 768721504, 3560215293, 12250746432, 144581799790, 2542575063630, 8955836934660, 55657973021431, 319349051391228, 1983548989621200, 7898257536096850

Offset: 0

Alois P. Heinz, Sep 28 2019

nonn

Sum of multinomials M(n; lambda), where lambda ranges over all integer partitions of n into distinct parts and one part is 2.

			a(2) = 1: 12.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 0.
a(5) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.

Alois P. Heinz, Table of n, a(n) for n = 0..697
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
Wikipedia, Partition of a set

Column k=2 of A327869.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$2, 0)-b(n$2, 2):
seq(a(n), n=0..29);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k] Binomial[n, i]]]];
a[n_] := b[n, n, 0] - b[n, n, 2];
a /@ Range[0, 29] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A327870 Row sums of A327869.

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Maple

Mathematica

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

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PARI

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Extensions

A327801 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A327884 Number T(n,k) of set partitions of [n] such that at least one of the block sizes is k or k=0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A328156 Number of set partitions of [2n] with distinct block sizes and one of the block sizes is n.

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Maple

Mathematica

Formula

A328155 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 3.

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Maple

Mathematica

A327876 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 1.

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Maple

Mathematica

Formula

A327881 Number of set partitions of [n] with distinct block sizes and one of the block sizes is 2.

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