A283877 -id:A283877

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A049311

Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

+10
133

1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908, 10725, 30825, 90156, 273234, 848355, 2714399, 8909057, 30042866, 103859678, 368075596, 1335537312, 4958599228, 18820993913, 72980867400, 288885080660, 1166541823566, 4802259167367, 20141650236664

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also the number of bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.

The EULERi transform (A056156) is also interesting.

a(n) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n. - Gus Wiseman, Mar 17 2017

LINKS

Aliaksandr Siarhei, Table of n, a(n) for n = 1..102

Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.

Peter J. Cameron, Problems on Permutation Groups, see Problem 3

Index entries for sequences related to binary matrices.

FORMULA

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform.

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; 1+x, 1+x^2, ...), where S_n X S_n is Cartesian product of symmetric groups S_n of degree n.

EXAMPLE

E.g. a(2) = 3: two ones in same row, two ones in same column, or neither.

a(3) = 6 is coefficient of x^3 in (1/36)*((1 + x)^9 + 6*(1 + x)^3*(1 + x^2)^3 + 8*(1 + x^3)^3 + 9*(1 + x)*(1 + x^2)^4 + 12*(1 + x^3)*(1 + x^6))=1 + x + 3*x^2 + 6*x^3 + 7*x^4 + 7*x^5 + 6*x^6 + 3*x^7 + x^8 + x^9.

There are a(3) = 6 binary matrices with 3 ones, with no zero rows or columns, up to row and column permutation:

[1 0 0] [1 1 0] [1 0] [1 1] [1 1 1] [1]

[0 1 0] [0 0 1] [1 0] [1 0] ....... [1].

[0 0 1] ....... [0 1] ............. [1]

Non-isomorphic representatives of the a(3)=6 set multipartitions are: ((123)), ((1)(23)), ((2)(12)), ((1)(1)(1)), ((1)(2)(2)), ((1)(2)(3)). - Gus Wiseman, Mar 17 2017

PROG

(PARI)

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}

a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 16 2023

CROSSREFS

Main diagonal of A321609.

Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.

Cf. A057149, A057150, A057151, A057152.

Cf. A034691, A056156, A089259, A116540, A283877.

KEYWORD

nonn,nice

AUTHOR

Peter J. Cameron

EXTENSIONS

More terms and formula from Vladeta Jovovic, Jul 29 2000

a(19)-a(28) from Max Alekseyev, Jul 22 2009

a(29)-a(102) from Aliaksandr Siarhei, Dec 13 2013

Name edited by Gus Wiseman, Dec 18 2018

STATUS

approved

A000612

Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2.
(Formerly M1712 N0677)

+10
131

1, 2, 6, 40, 1992, 18666624, 12813206169137152, 33758171486592987164087845043830784, 1435913805026242504952006868879460423834904914948818373264705576411070464

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Also number of nonisomorphic sets of nonempty subsets of an n-set.

Equivalently, number of nonisomorphic fillings of a Venn diagram of n sets. - Joerg Arndt, Mar 24 2020

Number of hypergraphs on n unlabeled nodes. - Charles R Greathouse IV, Apr 06 2021

REFERENCES

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 153.

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 Table 2.3.2. - Row 5.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..12

M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561.

M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561. [Annotated scanned copy]

Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, Hypergraph Motifs and Their Extensions Beyond Binary, arXiv:2310.15668 [cs.SI], 2023.

Peter H. van der Kamp, Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials, arXiv:2411.18264 [nlin.SI], 2024. See p. 4.

Wikipedia, Venn diagram

Index entries for sequences related to Boolean functions

FORMULA

a(n) = A003180(n)/2.

EXAMPLE

Non-isomorphic representatives of the a(2) = 6 set-systems are 0, {1}, {12}, {1}{2}, {1}{12}, {1}{2}{12}. - Gus Wiseman, Aug 07 2018

MAPLE

a:= n-> add(1/(p-> mul((c-> j^c*c!)(coeff(p, x, j)), j=1..degree(p)))(

add(x^i, i=l))*2^((w-> add(mul(2^igcd(t, l[i]), i=1..nops(l)),

t=1..w)/w)(ilcm(l[]))), l=combinat[partition](n))/2:

seq(a(n), n=0..9); # Alois P. Heinz, Aug 12 2019

MATHEMATICA

sysnorm[{}] := {}; sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]], sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}]], First[Sort[sysnorm[m, 1]]]]; sysnorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #>=aft&]}]}, Union@@(sysnorm[#, aft+1]&/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0, 1}], {par, First/@Position[mx, Max[mx]]}]])]];

Table[Length[Union[sysnorm/@Subsets[Rest[Subsets[Range[n]]]]]], {n, 4}] (* Gus Wiseman, Aug 07 2018 *)

a[n_] := Sum[1/Function[p, Product[Function[c, j^c*c!][Coefficient[p, x, j]], {j, 1, Exponent[p, x]}]][Total[x^l]]*2^(Function[w, Sum[Product[2^GCD[t, l[[i]]], {i, 1, Length[l]}], {t, 1, w}]/w][If[l=={}, 1, LCM @@ l]]), {l, IntegerPartitions[n]}]/2;

a /@ Range[0, 9] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz *)

CROSSREFS

a(n) = A003180(n)/2.

Cf. A007716, A055621, A058891, A283877, A300913, A306005, A317533, A317757.

KEYWORD

nonn,easy,nice,core,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Feb 23 2000

STATUS

approved

A367903

Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

+10
66

0, 0, 1, 67, 30997, 2147296425, 9223372036784737528, 170141183460469231731687303625772608225, 57896044618658097711785492504343953926634992332820282019728791606173188627779

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

LINKS

Table of n, a(n) for n=0..8.

Wikipedia, Axiom of choice.

FORMULA

a(n) + A367904(n) + A367772(n) = A058891(n+1) = 2^(2^n-1).

EXAMPLE

The a(2) = 1 set-system is {{1},{2},{1,2}}.

The a(3) = 67 set-systems have following 21 non-isomorphic representatives:

{{1},{2},{1,2}}

{{1},{2},{3},{1,2}}

{{1},{2},{3},{1,2,3}}

{{1},{2},{1,2},{1,3}}

{{1},{2},{1,2},{1,2,3}}

{{1},{2},{1,3},{2,3}}

{{1},{2},{1,3},{1,2,3}}

{{1},{1,2},{1,3},{2,3}}

{{1},{1,2},{1,3},{1,2,3}}

{{1},{1,2},{2,3},{1,2,3}}

{{1,2},{1,3},{2,3},{1,2,3}}

{{1},{2},{3},{1,2},{1,3}}

{{1},{2},{3},{1,2},{1,2,3}}

{{1},{2},{1,2},{1,3},{2,3}}

{{1},{2},{1,2},{1,3},{1,2,3}}

{{1},{2},{1,3},{2,3},{1,2,3}}

{{1},{1,2},{1,3},{2,3},{1,2,3}}

{{1},{2},{3},{1,2},{1,3},{2,3}}

{{1},{2},{3},{1,2},{1,3},{1,2,3}}

{{1},{2},{1,2},{1,3},{2,3},{1,2,3}}

{{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

MATHEMATICA

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n, 0, 3}]

CROSSREFS

Multisets of multisets of this type are ranked by A355529.

The version without singletons is A367769.

The version for simple graphs is A367867, covering A367868.

The version allowing empty edges is A367901.

The complement is A367902, without singletons A367770, ranks A367906.

For a unique choice (instead of none) we have A367904, ranks A367908.

These set-systems have ranks A367907.

An unlabeled version is A368094, for multiset partitions A368097.

A000372 counts antichains, covering A006126, nonempty A014466.

A003465 counts covering set-systems, unlabeled A055621.

A058891 counts set-systems, unlabeled A000612.

A059201 counts covering T_0 set-systems.

A323818 counts covering connected set-systems.

A326031 gives weight of the set-system with BII-number n.

Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 05 2023

EXTENSIONS

a(5)-a(8) from Christian Sievers, Jul 26 2024

STATUS

approved

A367902

Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

+10
64

1, 2, 7, 61, 1771, 187223, 70038280, 90111497503, 397783376192189

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

LINKS

Table of n, a(n) for n=0..8.

Wikipedia, Axiom of choice.

FORMULA

a(n) = A370636(2^n-1). - Alois P. Heinz, Mar 09 2024

EXAMPLE

The a(2) = 7 set-systems:

{}

{{1},{2}}

{{1},{1,2}}

{{2},{1,2}}

MATHEMATICA

Table[Length[Select[Subsets[Subsets[Range[n]]], Select[Tuples[#], UnsameQ@@#&]!={}&]], {n, 0, 3}]

CROSSREFS

The version for simple graphs is A133686, covering A367869.

The version without singletons is A367770.

The complement allowing empty edges is A367901.

The complement is A367903, without singletons A367769, ranks A367907.

For a unique choice we have A367904, ranks A367908.

These set-systems have ranks A367906.

A000372 counts antichains, covering A006126, nonempty A014466.

A003465 counts covering set-systems, unlabeled A055621.

A058891 counts set-systems, unlabeled A000612.

A059201 counts covering T_0 set-systems.

A323818 counts covering connected set-systems.

A326031 gives weight of the set-system with BII-number n.

Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A370636.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 05 2023

EXTENSIONS

a(6)-a(8) from Christian Sievers, Jul 25 2024

STATUS

approved

A306017

Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

+10
58

1, 1, 4, 6, 17, 14, 66, 30, 189, 222, 550, 112, 4696, 202, 5612, 30914, 63219, 594, 453125, 980, 3602695, 5914580, 1169348, 2510, 299083307, 232988061, 23248212, 2669116433, 14829762423, 9130, 170677509317, 13684, 1724710753084, 2199418340875, 14184712185, 38316098104262

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.

Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018

From Gus Wiseman, Oct 11 2018: (Start)

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:

{{1},{1,1,1}}

{{1},{1,2,2}}

{{1},{2,3,4}}

{{1,1},{1,1}}

{{1,1},{2,2}}

{{1,2},{1,2}}

{{1,2},{3,4}}

{{1},{1},{1,1}}

{{1},{1},{2,2}}

{{1},{2},{1,2}}

{{1},{2},{3,4}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{3},{4}}

(End)

LINKS

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

FORMULA

For p prime, a(p) = 2*A000041(p).

a(n) = Sum_{d|n} A331485(n/d, d). - Andrew Howroyd, Feb 09 2020

EXAMPLE

Non-isomorphic representatives of the a(4) = 17 multiset partitions:

{{1,1},{1,1}}

{{1,1},{2,2}}

{{1,2},{1,2}}

{{1,2},{2,2}}

{{1,2},{3,3}}

{{1,2},{3,4}}

{{1,3},{2,3}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{2},{2}}

{{1},{2},{3},{3}}

{{1},{2},{3},{4}}

MATHEMATICA

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];

K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];

RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];

a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]];

Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 07 2019, after Andrew Howroyd *)

PROG

(PARI) \\ See A318951 for RowSumMats.

a(n)={sumdiv(n, d, RowSumMats(n/d, n, d))} \\ Andrew Howroyd, Sep 05 2018

CROSSREFS

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.

Cf. A064573, A279787, A305551, A319616, A319056, A331485.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 17 2018

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018

STATUS

approved

A300913

Number of non-isomorphic connected set-systems of weight n.

+10
55

1, 1, 1, 2, 4, 7, 18, 37, 96, 239, 658, 1810, 5358, 16057, 50373, 161811, 536964, 1826151, 6380481, 22822280, 83587920, 312954111, 1197178941, 4674642341, 18620255306, 75606404857, 312763294254, 1317356836235, 5646694922172, 24618969819915, 109125629486233, 491554330852608

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The weight of a set-system is the sum of cardinalities of the sets. Weight is generally not the same as number of vertices.

LINKS

Jean-François Alcover, Table of n, a(n) for n = 0..50 [using Andrew Howroyd's b-file for A283877]

FORMULA

Inverse Euler transform of A283877.

EXAMPLE

Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 set systems:

1: {{1}}

2: {{1,2}}

3: {{1,2,3}}

{{2},{1,2}}

4: {{1,2,3,4}}

{{3},{1,2,3}}

{{1,3},{2,3}}

{{1},{2},{1,2}}

5: {{1,2,3,4,5}}

{{4},{1,2,3,4}}

{{1,4},{2,3,4}}

{{2,3},{1,2,3}}

{{2},{3},{1,2,3}}

{{2},{1,3},{2,3}}

{{3},{1,3},{2,3}}

Non-isomorphic representatives of the a(6) = 18 connected set-systems:

{{5},{1,2,3,4,5}}

{{1,5},{2,3,4,5}}

{{3,4},{1,2,3,4}}

{{1,2,5},{3,4,5}}

{{1,3,4},{2,3,4}}

{{1},{1,4},{2,3,4}}

{{1},{2,3},{1,2,3}}

{{3},{4},{1,2,3,4}}

{{3},{1,4},{2,3,4}}

{{3},{2,3},{1,2,3}}

{{4},{1,4},{2,3,4}}

{{1,2},{1,3},{2,3}}

{{1,3},{2,4},{3,4}}

{{1,4},{2,4},{3,4}}

{{1},{2},{3},{1,2,3}}

{{1},{2},{1,3},{2,3}}

{{2},{3},{1,3},{2,3}}

MATHEMATICA

A283877 = Import["https://oeis.org/A283877/b283877.txt", "Table"][[All, 2]];

(* EulerInvTransform is defined in A022562 *)

{1} ~Join~ EulerInvTransform[A283877 // Rest] (* Jean-François Alcover, Nov 07 2019, updated Mar 17 2020 *)

CROSSREFS

Cf. A007716, A055621, A283877, A293606, A293607.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 19 2018

EXTENSIONS

a(11)-a(31) from Jean-François Alcover, Nov 07 2019

STATUS

approved

A319056

Number of non-isomorphic multiset partitions of weight n in which (1) all parts have the same size and (2) each vertex appears the same number of times.

+10
47

1, 1, 4, 4, 10, 4, 21, 4, 26, 13, 28, 4, 128, 4, 39, 84, 150, 4, 358, 4, 956, 513, 86, 4, 12549, 1864, 134, 9582, 52366, 4, 301086, 4, 1042038, 407140, 336, 4690369, 61738312, 4, 532, 28011397, 2674943885, 4, 819150246, 4, 54904825372, 65666759973, 1303, 4, 4319823776760

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(p) = 4 for p prime. - Charlie Neder, Oct 15 2018

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..75

EXAMPLE

Non-isomorphic representatives of the a(1) = 1 through a(6) = 21 multiset partitions:

(1) (11) (111) (1111) (11111) (111111)

(12) (123) (1122) (12345) (111222)

(1)(1) (1)(1)(1) (1234) (1)(1)(1)(1)(1) (112233)

(1)(2) (1)(2)(3) (11)(11) (1)(2)(3)(4)(5) (123456)

(11)(22) (111)(111)

(12)(12) (111)(222)

(12)(34) (112)(122)

(1)(1)(1)(1) (112)(233)

(1)(1)(2)(2) (123)(123)

(1)(2)(3)(4) (123)(456)

(11)(11)(11)

(11)(12)(22)

(11)(22)(33)

(11)(23)(23)

(12)(12)(12)

(12)(13)(23)

(12)(34)(56)

(1)(1)(1)(1)(1)(1)

(1)(1)(1)(2)(2)(2)

(1)(1)(2)(2)(3)(3)

(1)(2)(3)(4)(5)(6)

CROSSREFS

Row sums of A322789.

Cf. A007716, A049311, A064573, A279787, A283877, A306017, A316980, A316981, A316983, A319616.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 10 2018

EXTENSIONS

Terms a(12) and beyond from Andrew Howroyd, Feb 03 2022

STATUS

approved

A293606

Number of unlabeled antichains of weight n.

+10
46

1, 1, 2, 3, 6, 9, 20, 33, 72, 139

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

An antichain is a finite set of finite nonempty sets, none of which is a subset of any other. The weight of an antichain is the sum of cardinalities of its elements.

From Gus Wiseman, Aug 15 2019: (Start)

Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n where every vertex is the unique common element of some subset of the edges. For example, the a(1) = 1 through a(6) = 20 set multipartitions are:

{1} {1}{1} {1}{1}{1} {1}{2}{12} {1}{2}{2}{12} {12}{13}{23}

{1}{2} {1}{2}{2} {1}{1}{1}{1} {1}{2}{3}{23} {1}{2}{12}{12}

{1}{2}{3} {1}{1}{2}{2} {1}{1}{1}{1}{1} {1}{2}{13}{23}

{1}{2}{2}{2} {1}{1}{2}{2}{2} {1}{2}{3}{123}

{1}{2}{3}{3} {1}{2}{2}{2}{2} {1}{1}{2}{2}{12}

{1}{2}{3}{4} {1}{2}{2}{3}{3} {1}{1}{2}{3}{23}

{1}{2}{3}{3}{3} {1}{2}{2}{2}{12}

{1}{2}{3}{4}{4} {1}{2}{3}{3}{23}

{1}{2}{3}{4}{5} {1}{2}{3}{4}{34}

{1}{1}{1}{1}{1}{1}

{1}{1}{1}{2}{2}{2}

{1}{1}{2}{2}{2}{2}

{1}{1}{2}{2}{3}{3}

{1}{2}{2}{2}{2}{2}

{1}{2}{2}{3}{3}{3}

{1}{2}{3}{3}{3}{3}

{1}{2}{3}{3}{4}{4}

{1}{2}{3}{4}{4}{4}

{1}{2}{3}{4}{5}{5}

{1}{2}{3}{4}{5}{6}

(End)

LINKS

Table of n, a(n) for n=0..9.

FORMULA

Euler transform of A293607.

EXAMPLE

Non-isomorphic representatives of the a(5) = 9 antichains are:

((12345)),

((1)(2345)), ((12)(134)), ((12)(345)),

((1)(2)(345)), ((1)(23)(45)), ((2)(13)(14)),

((1)(2)(3)(45)),

((1)(2)(3)(4)(5)).

CROSSREFS

Cf. A006126, A006602, A007411, A007716, A048143, A049311, A283877, A293607.

Cf. A000372, A000612, A003182, A014466, A055621, A293993, A326704, A326972.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Oct 13 2017

STATUS

approved

A302545

Number of non-isomorphic multiset partitions of weight n with no singletons.

+10
46

1, 0, 2, 3, 12, 23, 84, 204, 682, 1977, 6546, 21003, 72038, 248055, 888771, 3240578, 12152775, 46527471, 182339441, 729405164, 2979121279, 12407308136, 52670355242, 227725915268, 1002285274515, 4487915293698, 20434064295155, 94559526596293, 444527730210294, 2122005930659752

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. A singleton is a multiset of size 1. The weight of a multiset partition is the sum of sizes of its elements. Weight is generally not the same as number of vertices.

Also non-isomorphic multiset partitions of weight n with no endpoints, where an endpoint is a vertex appearing only once (degree 1). For example, non-isomorphic representations of the a(4) = 12 multiset partitions are:

{{1},{1,1,1}}

{{1},{1,2,2}}

{{1,1},{1,1}}

{{1,1},{2,2}}

{{1,2},{1,2}}

{{1},{1},{1,1}}

{{1},{1},{2,2}}

{{1},{2},{1,2}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

LINKS

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

EXAMPLE

The a(4) = 12 multiset partitions:

{{1,1},{1,1}}

{{1,1},{2,2}}

{{1,2},{1,2}}

{{1,2},{2,2}}

{{1,2},{3,3}}

{{1,2},{3,4}}

{{1,3},{2,3}}

PROG

(PARI) \\ compare with similar program for A007716.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t)) + O(x*x^k), -k)}

a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 15 2023

CROSSREFS

The set-system version is A330054 (no endpoints) or A306005 (no singletons).

Non-isomorphic multiset partitions are A007716.

Set-systems with no singletons are A016031.

Cf. A049311, A283877, A293606, A293607, A306008, A317533, A317794, A317795, A320665, A330053, A330055, A330058.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 20 2018

EXTENSIONS

Extended by Gus Wiseman, Dec 09 2019

Terms a(11) and beyond from Andrew Howroyd, Jan 15 2023

STATUS

approved

A306021

Number of set-systems spanning {1,...,n} in which all sets have the same size.

+10
43

1, 1, 2, 6, 54, 1754, 1102746, 68715913086, 1180735735356265746734, 170141183460507906731293351306656207090, 7237005577335553223087828975127304177495735363998991435497132232365910414322

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(n) is the number of labeled uniform hypergraphs spanning n vertices. - Andrew Howroyd, Jan 16 2024

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..14

FORMULA

a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)).

Inverse binomial transform of A306020. - Andrew Howroyd, Jan 16 2024

EXAMPLE

The a(3) = 6 set-systems in which all sets have the same size:

{{1}, {2}, {3}}

{{1,2}, {1,3}}

{{1,2}, {2,3}}

{{1,3}, {2,3}}

{{1,2}, {1,3}, {2,3}}

MATHEMATICA

Table[Sum[(-1)^(n-k)*Binomial[n, k]*(1+Sum[2^Binomial[k, d]-1, {d, k}]), {k, 0, n}], {n, 12}]

PROG

(PARI) a(n) = if(n==0, 1, sum(k=0, n, sum(d=0, n, (-1)^(n-d)*binomial(n, d)*2^binomial(d, k)))) \\ Andrew Howroyd, Jan 16 2024

CROSSREFS

Row sums of A299471.

The unlabeled version is A301481.

The connected version is A299353.

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306019, A306020.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 17 2018

STATUS

approved

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