A038041 -id:A038041 - cp's OEIS frontend

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

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 *)

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}]

A361866 Number of set partitions of {1..n} with block-means summing to an integer.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 75, 267, 1119, 4965, 22694, 117090, 670621, 3866503, 24113829, 161085223, 1120025702, 8121648620, 62083083115, 492273775141, 4074919882483

Offset: 0

Gus Wiseman, Apr 04 2023

			The a(1) = 1 through a(4) = 8 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{13}{24}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
The set partition y = {{1,2},{3,4}} has block-means {3/2,7/2}, with sum 5, so y is counted under a(4).

For mean instead of sum we have A361865, for median A361864.
For median instead of mean we have A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions with integer mean block-size.
A327475 counts subsets with integer mean, median A000975.
A327481 counts subsets by mean, median A013580.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A067659, A326515, A326516, A326521.

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

a(14)-a(20) from Christian Sievers, May 12 2025

A073705 a(n) = Sum_{ d divides n } (n/d)^(2d).

Original entry on oeis.org

1, 5, 10, 33, 26, 182, 50, 577, 811, 1750, 122, 16194, 170, 18982, 74900, 135425, 290, 847127, 362, 2498178, 4901060, 4209430, 530, 78564226, 9766251, 67138102, 387952660, 542674914, 842, 4866184552, 962, 8606778369, 31382832260, 17179953862, 6385992100, 422091411267, 1370, 274878038710

Offset: 1

Paul D. Hanna, Aug 04 2002

a(n) is the number of linear partitions of the linearly ordered set [n] = {1,2,...,n} with blocks of the same size, where each block has two element marked (possibly equal). For instance, for n = 3, we have the following 10 linear partitions (where the marked elements are denoted by a and b, or by X when they coincide):

(X)(X)(X), (ab3), (a2b), (1ab), (ba3), (b2a), (1ba), (X23), (1X3), (12X). - Emanuele Munarini, Feb 03 2014

			a(10) = (10/1)^(2*1) +(10/2)^(2*2) +(10/5)^(2*5) +(10/10)^(2*10) = 1750 because positive divisors of 10 are 1, 2, 5, 10.

Seiichi Manyama, Table of n, a(n) for n = 1..3143

Cf. A038041, A055225, A151954, A236696.

Table[Total[Quotient[n, x = Divisors[n]]^(2*x)], {n, 34}] (* Jayanta Basu, Jul 08 2013 *)

a(n):= lsum(d^(2*n/d),d,listify(divisors(n)));
makelist(a(n),n,1,40); /* Emanuele Munarini , Feb 03 2014 */

a(n)=sumdiv(n, d, (d)^(2*n/d) );  /* Joerg Arndt, Oct 07 2012 */

G.f.: Sum_{n>=1} -log(1 - (n^2)*x^n)/n = Sum_{n>=1} a(n) x^n/n.

G.f.: Sum_{k>=1} k^2*x^k/(1-k^2*x^k). - Benoit Cloitre, Apr 21 2003

Corrected a(14) and inserted missing a(16) by Jayanta Basu, Jul 08 2013

A208437 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} that have exactly k distinct block sizes.

Original entry on oeis.org

1, 2, 2, 3, 5, 10, 2, 50, 27, 116, 60, 2, 560, 315, 142, 1730, 2268, 282, 6123, 14742, 1073, 30122, 72180, 12600, 2, 116908, 464640, 97020, 32034, 507277, 2676366, 997920, 2, 2492737, 16400098, 8751600, 136853, 15328119, 94209206, 81225144, 1527528, 56182092, 673282610, 614128515, 37837800

Offset: 1

Geoffrey Critzer, Feb 26 2012

Column 1 = A038041.
Column 2 = A088142.
Column 3 = A133118.
Row sums = A000110 (Bell numbers).
Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number). - Franklin T. Adams-Watters, Feb 26 2012

			:    1;
:    2;
:    2,      3;
:    5,     10;
:    2,     50;
:   27,    116,     60;
:    2,    560,    315;
:  142,   1730,   2268;
:  282,   6123,  14742;
: 1073,  30122,  72180,   12600;

Alois P. Heinz, Rows n = 1..220, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 180.

Cf. A022915, A116608, A218868.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Aug 21 2014

nn = 15; p = Product[1 + y (Exp[x^i/i!] - 1), {i, 1, nn}];f[list_] := Select[list, # > 0 &];
Map[f, Drop[ Range[0, nn]! CoefficientList[Series[p, {x, 0, nn}], {x, y}], 1]] // Flatten

E.g.f.: Product_{i>=1} 1 + y *(exp(x^i/i!)-1).

T(n*(n+1)/2,n) = A022915(n). - Alois P. Heinz, Apr 08 2016

A301481 Number of unlabeled uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 2, 4, 12, 58, 2381, 14026281, 29284932065996445, 468863491068204425232922367150021, 1994324729204021501147398087008429476673379600542622915802043462326345

Offset: 0

Gus Wiseman, Jun 19 2018

nonn

A hypergraph is uniform if all edges have the same size.

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

Alois P. Heinz, Table of n, a(n) for n = 0..14 (first 13 terms from Andrew Howroyd)

Row sums of A301922.

Cf. A003465, A038041, A055621, A298422, A301920, A306017-A306021.

\\ see A301922 for U(n,k).
a(n)={ if(n==0, 1, sum(k=1, n, U(n,k)-U(n-1,k))) } \\ Andrew Howroyd, Aug 10 2019

Terms a(6) and beyond from Andrew Howroyd, Aug 09 2019

A320444 Number of uniform hypertrees spanning n vertices.

Original entry on oeis.org

1, 1, 1, 4, 17, 141, 1297, 17683, 262145, 4861405, 100112001, 2371816701, 61917364225, 1796326510993, 56693912375297, 1947734359001551, 72059082110369793, 2863257607266475419, 121439531096594251777, 5480987217944109919765, 262144000000000000000001

Offset: 0

Gus Wiseman, Jan 09 2019

nonn

The density of a hypergraph is the sum of sizes of its edges minus the number of edges minus the number of vertices. A hypertree is a connected hypergraph of density -1. A hypergraph is uniform if its edges all have the same size. The span of a hypergraph is the union of its edges.

			Non-isomorphic representatives of the 5 unlabeled uniform hypertrees on 5 vertices and their multiplicities in the labeled case, which add up to a(5) = 141:
   5 X {{1,5},{2,5},{3,5},{4,5}}
  60 X {{1,4},{2,5},{3,5},{4,5}}
  60 X {{1,3},{2,4},{3,5},{4,5}}
  15 X {{1,2,5},{3,4,5}}
   1 X {{1,2,3,4,5}}

Robert Israel, Table of n, a(n) for n = 0..387

Row sums of A326374.

Cf. A000272, A030019, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

f:= proc(n) local d; add((n-1)!/(d! * ((n-1)/d)!) * (n/d)^((n-1)/d - 1), d = numtheory:-divisors(n-1)); end proc:
f(0):= 1: f(1):= 1:
map(f, [$0..25]); # Robert Israel, Jan 10 2019

Table[Sum[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{d,Divisors[n]}],{n,10}]

a(n) = if (n<2, 1, n--; sumdiv(n, d, n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1))); \\ Michel Marcus, Jan 10 2019

a(n + 1) = Sum_{d|n} n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1).

a(p prime) = 1 + (p + 1)^(p - 1).

A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 31, 34, 45, 51, 63, 72, 88, 97, 120, 128, 158, 174, 201, 222, 264, 287, 333, 359, 416, 441, 518, 557, 631, 684, 770, 833, 954, 1017, 1141, 1222, 1378, 1475, 1643, 1755, 1939, 2097, 2327, 2471, 2758, 2928, 3233, 3470, 3813, 4085

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 18 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (52)       (44)
             (111)  (31)    (41)     (42)      (61)       (53)
                    (211)   (221)    (51)      (331)      (71)
                    (1111)  (311)    (222)     (421)      (422)
                            (2111)   (321)     (511)      (521)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (3211)     (2222)
                                     (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
Missing from the list for n = 7 through 9:
  (43)   (62)    (54)
  (322)  (332)   (63)
         (431)   (432)
         (3221)  (522)
                 (621)
                 (3222)
                 (3321)
                 (4311)
                 (32211)

Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.

Table[Length[Select[IntegerPartitions[n],SameQ@@Last/@FactorInteger[Times@@#]&]],{n,30}]

A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.

Original entry on oeis.org

1, 1, 2, 5, 11, 23, 49, 103, 214, 434, 874, 1738, 3443, 6765, 13193, 25512, 48957, 93267, 176595, 332550, 622957, 1161230, 2153710, 3974809, 7299707, 13343290, 24280924, 43999100, 79412942, 142792535, 255826836, 456735456, 812627069, 1440971069, 2546729830

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 11 sequences:
  (1)  (2)   (3)      (4)
       (11)  (21)     (22)
             (111)    (31)
             (1)(11)  (211)
             (11)(1)  (1111)
                      (11)(2)
                      (1)(21)
                      (2)(11)
                      (21)(1)
                      (1)(111)
                      (111)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of set partitions is A007837.
This is the case of A055887 with all distinct lengths.
For distinct sums instead of lengths we have A336342.
The case of twice-partitions is A358830.
The unordered version is A358836.
The version for constant instead of distinct lengths is A358905.
A000041 counts integer partitions, strict A000009.
A063834 counts twice-partitions.
A141199 counts sequences of partitions with weakly decreasing lengths.
Cf. A000219, A001970, A038041, A060642, A218482, A271619, A319066, A358831, A358901, A358906, A358908.

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@Length/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); [subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, 1 + y*polcoef(g, k, y) + O(x*x^n)))]} \\ Andrew Howroyd, Dec 30 2022

Terms a(16) and beyond from Andrew Howroyd, Dec 30 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A361866 Number of set partitions of {1..n} with block-means summing to an integer.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A073705 a(n) = Sum_{ d divides n } (n/d)^(2d).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

Extensions

A208437 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} that have exactly k distinct block sizes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A301481 Number of unlabeled uniform hypergraphs spanning n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A320444 Number of uniform hypertrees spanning n vertices.

Views

Author

Keywords