A007837 -id:A007837 - cp's OEIS frontend

A185895 Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!)....

1, -1, -1, 2, 3, 14, -40, -43, -357, -1762, 8004, 13067, 78540, 492439, 3932305, -26867293, -44643557, -363632466, -1729625764, -15939972937, -145669871232, 1488599170613, 3515325612655, 26765194180353, 151925998229148

Offset: 0

Michael Somos, Feb 05 2011

sign

From Peter Bala, Mar 17 2022: (Start)
Conjectures: 1) a(n) differs in sign from a(n-1) iff n is a triangular number (checked up to n = 1225 = (50*51)/2)
2) The same property holds for the coefficients of A(x)^2, the square of the o.g.f. A(x) =  1 - x - x^2 + 2*x^3 + 3*x^4 + ... : A(x)^2 = 1 - 2*x - x^2 + 6*x^3 + 3*x^4 + 18*x^5 - 110*x^6 - 22*x^7 - 483*x^8 - 2800*x^9 + 20030*x^10 + ....
3) 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. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..300

Cf. A005651, A007837, A168268.

{a(n) = if( n<0, 0, n! * polcoeff( prod( k=1, n, 1 - x^k / k!, 1 + x * O(x^n)), n))}

{a(n)=if(n<0,0,if(n==0,1,sum(k=1,n,(n-1)!/(n-k)!*a(n-k)*sumdiv(k,d,-d*d!^(-k/d)))))} [Hanna]

E.g.f.: Product_{k>0} (1 - x^k/k!).
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 [cf. Vladeta Jovovic's formula in A007837].
E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

A275313 Number of set partitions of [n] where adjacent blocks differ in size.

Original entry on oeis.org

1, 1, 1, 4, 7, 23, 100, 333, 1443, 6910, 36035, 186958, 1095251, 6620976, 42151463, 290483173, 2030271491, 15044953241, 116044969497, 930056879535, 7749440529803, 66931578540965, 597728811956244, 5511695171795434, 52578231393128128, 515775207055816041

Offset: 0

Alois P. Heinz, Jul 22 2016

nonn

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 7: 1234, 123|4, 124|3, 134|2, 1|234, 1|23|4, 1|24|3.
a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|2|45, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34.

Alois P. Heinz, Table of n, a(n) for n = 0..580
Wikipedia, Partition of a set

Cf. A007837, A038041, A275309, A275310, A275311, A275312, A275679, A280275, A286072.

b:= proc(n, i) option remember; `if`(n=0, 1, add(`if`(i=j, 0,
      b(n-j, `if`(j>n-j, 0, j))*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..35);

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[If[i==j, 0, b[n-j, If[j>n-j, 0, j]]* Binomial[n-1, j-1]], {j, 1, n}]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 4, 3, 3, 1, 5, 4, 0, 4, 1, 16, 5, 10, 10, 5, 1, 82, 66, 75, 60, 15, 6, 1, 169, 112, 126, 35, 140, 21, 7, 1, 541, 456, 196, 336, 280, 224, 28, 8, 1, 2272, 765, 1548, 1848, 1386, 630, 336, 36, 9, 1, 17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1

Offset: 0

Alois P. Heinz, Sep 28 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.

Number T(n,k) of set partitions of [n] with distinct block sizes and one of the block sizes is k. T(5,3) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.

			Triangle T(n,k) begins:
      1;
      1,     1;
      1,     0,     1;
      4,     3,     3,     1;
      5,     4,     0,     4,     1;
     16,     5,    10,    10,     5,    1;
     82,    66,    75,    60,    15,    6,    1;
    169,   112,   126,    35,   140,   21,    7,   1;
    541,   456,   196,   336,   280,  224,   28,   8,  1;
   2272,   765,  1548,  1848,  1386,  630,  336,  36,  9,  1;
  17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1;
  ...

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

Columns k=0-3 give: A007837, A327876, A327881, A328155.
Row sums give A327870.
T(2n,n) gives A328156.
Cf. A327801, A327884.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0),
             l=select(x-> nops(x)=nops({x[]}) and
             (k=0 or k in x), partition(n))):
seq(seq(T(n, k), k=0..n), n=0..11);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..11);

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!]]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, from 2nd Maple program *)

A032299 "EFJ" (unordered, size, labeled) transform of 1,2,3,4,...

Original entry on oeis.org

1, 1, 2, 9, 16, 85, 516, 1519, 6280, 45441, 431740, 1394371, 8370924, 43960657, 459099018, 6135631545, 23813007376, 150537761905, 1029390040764, 7519458731131, 101693768415220, 1909742186139921, 8269148260309882, 60924484457661793, 417027498430063800

Offset: 0

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
C. G. Bower, Transforms (2)

Cf. A007837, A076900.

mmax = 25;
egf = Product[1 + x^m/(m - 1)!, {m, 1, mmax}] + O[x]^mmax;
CoefficientList[egf, x] * Range[0, mmax - 1]! (* Jean-François Alcover, Sep 23 2019 *)

seq(n)={Vec(serlaplace(prod(k=1, n, 1 + k*x^k/k! + O(x*x^n))))} \\ Andrew Howroyd, Sep 11 2018

E.g.f.: Product_{m>0} (1+x^m/(m-1)!). - Vladeta Jovovic, Nov 26 2002

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*((j - 1)!)^k)). - Ilya Gutkovskiy, Sep 13 2018

a(0)=1 prepended and terms a(23) and beyond from Andrew Howroyd, Sep 11 2018

A275309 Number of set partitions of [n] with decreasing block sizes.

Original entry on oeis.org

1, 1, 1, 3, 4, 11, 36, 82, 239, 821, 3742, 10328, 42934, 156070, 729249, 4025361, 15032099, 68746675, 334541624, 1645575386, 9104991312, 65010298257, 282768687257, 1616844660914, 8660050947383, 53262316928024, 309119883729116, 2185141720645817

Offset: 0

Alois P. Heinz, Jul 22 2016

nonn

			a(3) = 3: 123, 12|3, 13|2.
a(4) = 4: 1234, 123|4, 124|3, 134|2.
a(5) = 11: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 1345|2, 134|25, 135|24, 145|23.

Alois P. Heinz, Table of n, a(n) for n = 0..717
Wikipedia, Partition of a set

Cf. A007837, A038041, A275310, A275311, A275312, A275313, A286073.

b:= proc(n, i) option remember;
      `if`(n>i*(i+1)/2, 0, `if`(n=0, 1, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i-1)*binomial(n-1, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);

b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, b[n, i - 1] +  If[i > n, 0, b[n - i, i - 1]*Binomial[n - 1, i - 1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 21 2017, translated from Maple *)

A275310 Number of set partitions of [n] with nonincreasing block sizes.

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 102, 346, 1353, 5444, 24170, 110082, 546075, 2777828, 15099359, 84491723, 499665713, 3035284304, 19375261490, 126821116410, 866293979945, 6072753348997, 44193947169228, 329387416656794, 2542173092336648, 20069525888319293

Offset: 0

Alois P. Heinz, Jul 22 2016

nonn

			a(3) = 4: 123, 12|3, 13|2, 1|2|3.
a(4) = 11: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 14|2|3, 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..612
Wikipedia, Partition of a set

Cf. A007837, A038041, A275309, A275311, A275312, A275313, A286071.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-j, j)*binomial(n-1, j-1), j=1..min(n, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j]*Binomial[n-1, j-1], {j, 1, Min[n, i]}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

A275311 Number of set partitions of [n] with nondecreasing block sizes.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 43, 89, 363, 1096, 4349, 14575, 77166, 265648, 1369284, 6700177, 33526541, 162825946, 1034556673, 5157939218, 33054650345, 206612195885, 1244742654646, 8071979804457, 62003987375957, 381323590616995, 2827411772791596, 22061592185044910

Offset: 0

Alois P. Heinz, Jul 22 2016

nonn

			a(3) = 3: 123, 1|23, 1|2|3.
a(4) = 7: 1234, 12|34, 13|24, 14|23, 1|234, 1|2|34, 1|2|3|4.
a(5) = 12: 12345, 12|345, 13|245, 14|235, 15|234, 1|2345, 1|23|45, 1|24|35, 1|25|34, 1|2|345, 1|2|3|45, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..617
Wikipedia, Partition of a set

Cf. A007837, A038041, A275309, A275310, A275312, A275313, A286074.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, j)*binomial(n-1, j-1), j=i..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..35);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, j]*Binomial[n-1, j-1], {j, i, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 22 2017, translated from Maple *)

A275312 Number of set partitions of [n] with increasing block sizes.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 11, 28, 51, 242, 532, 1545, 6188, 16592, 86940, 302909, 967523, 3808673, 23029861, 71772352, 484629531, 1840886853, 9376324526, 37878035106, 204542429832, 1458360522892, 6241489795503, 45783932444672, 211848342780210, 1137580874772724

Offset: 0

Alois P. Heinz, Jul 22 2016

nonn

			a(4) = 2: 1234, 1|234.
a(5) = 6: 12345, 12|345, 13|245, 14|235, 15|234, 1|2345.
a(6) = 11: 123456, 12|3456, 13|2456, 14|2356, 15|2346, 16|2345, 1|23456, 1|23|456, 1|24|356, 1|25|346, 1|26|345.

Alois P. Heinz, Table of n, a(n) for n = 0..728
Wikipedia, Partition of a set

Cf. A007837, A038041, A275309, A275310, A275311, A275313, A286075.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i+1)*binomial(n-1, i-1)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..35);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i+1] + b[n-i, i+1] * Binomial[n-1, i-1]]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 22 2017, translated from Maple *)

A326026 Number of non-isomorphic multiset partitions of weight n where each part has a different length.

Original entry on oeis.org

1, 1, 2, 7, 12, 35, 111, 247, 624, 1843, 6717, 15020, 46847, 124808, 412577, 1658973, 4217546, 12997734, 40786810, 126971940, 437063393, 2106317043, 5499108365, 19037901867, 59939925812, 210338815573, 683526043801, 2741350650705, 14848209030691, 41533835240731, 151548411269815

Offset: 0

Gus Wiseman, Jul 13 2019

nonn

The number of non-isomorphic multiset partitions of weight n is A007716(n).

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{1},{1,1}}  {{1,2,3,3}}
                  {{1},{2,2}}  {{1,2,3,4}}
                  {{1},{2,3}}  {{1},{1,1,1}}
                  {{2},{1,2}}  {{1},{1,2,2}}
                               {{1},{2,2,2}}
                               {{1},{2,3,3}}
                               {{1},{2,3,4}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A332260.

Cf. A007716, A007837, A303546, A306017, A316980, A316983, A319560, A326514, A326517, A326533.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p,n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); polcoef(prod(k=1, #u, 1 + u[k]*x^k + O(x*x^n)), n)/prod(i=1, #v, i^v[i]*v[i]!)}
a(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Feb 08 2020

Terms a(11) and beyond from Andrew Howroyd, Feb 08 2020

A361864 Number of set partitions of {1..n} whose block-medians have integer median.

Original entry on oeis.org

1, 0, 3, 6, 30, 96, 461, 2000, 10727, 57092, 342348

Offset: 1

Gus Wiseman, Apr 04 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 1 through a(4) = 6 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}
            {{13}{2}}    {{123}{4}}
            {{1}{2}{3}}  {{1}{2}{34}}
                         {{12}{3}{4}}
                         {{1}{24}{3}}
                         {{13}{2}{4}}
The set partition {{1,2},{3},{4}} has block-medians {3/2,3,4}, with median 3, so is counted under a(4).

For mean instead of median we have A361865.
For sum instead of outer median we have A361911, means A361866.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A308037 counts set partitions with integer average block-size.
A325347 counts partitions w/ integer median, complement A307683.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A359893, A359907, A361801.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Median[Median/@#]]&]],{n,6}]

cp's OEIS Frontend

A185895 Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!)....

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PARI

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A275313 Number of set partitions of [n] where adjacent blocks differ in size.

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Maple

Mathematica

A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Mathematica

A032299 "EFJ" (unordered, size, labeled) transform of 1,2,3,4,...

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Mathematica

PARI

Formula

Extensions

A275309 Number of set partitions of [n] with decreasing block sizes.

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Maple

Mathematica

A275310 Number of set partitions of [n] with nonincreasing block sizes.

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Maple

Mathematica

A275311 Number of set partitions of [n] with nondecreasing block sizes.

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Maple

Mathematica

A275312 Number of set partitions of [n] with increasing block sizes.

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