Search: a283877 -id:a283877
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A049311
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Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.
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+10
133
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
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FORMULA
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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.
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A000612
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Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2.
(Formerly M1712 N0677)
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+10
131
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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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:
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MATHEMATICA
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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;
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CROSSREFS
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KEYWORD
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nonn,easy,nice,core
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A367903
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Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.
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+10
67
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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}}
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MATHEMATICA
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Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n, 0, 3}]
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CROSSREFS
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Multisets of multisets of this type are ranked by A355529.
The version without singletons is A367769.
The version allowing empty edges is A367901.
These set-systems have ranks A367907.
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.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A367902
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Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.
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+10
64
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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The a(2) = 7 set-systems:
{}
{{1}}
{{2}}
{{1,2}}
{{1},{2}}
{{1},{1,2}}
{{2},{1,2}}
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n]]], Select[Tuples[#], UnsameQ@@#&]!={}&]], {n, 0, 3}]
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CROSSREFS
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The version without singletons is A367770.
The complement allowing empty edges is A367901.
These set-systems have ranks A367906.
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.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A306017
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Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.
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+10
58
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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
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OFFSET
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0,3
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COMMENTS
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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
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},{2,3,4}}
{{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)
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LINKS
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FORMULA
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EXAMPLE
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Non-isomorphic representatives of the a(4) = 17 multiset partitions:
{{1,1,1,1}}
{{1,1,2,2}}
{{1,2,2,2}}
{{1,2,3,3}}
{{1,2,3,4}}
{{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}}
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MATHEMATICA
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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]}]]];
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PROG
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(PARI) \\ See A318951 for RowSumMats.
a(n)={sumdiv(n, d, RowSumMats(n/d, n, d))} \\ Andrew Howroyd, Sep 05 2018
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CROSSREFS
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Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A300913
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Number of non-isomorphic connected set-systems of weight n.
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+10
55
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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
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OFFSET
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0,4
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COMMENTS
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The weight of a set-system is the sum of cardinalities of the sets. Weight is generally not the same as number of vertices.
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LINKS
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FORMULA
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Inverse Euler transform of A283877.
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EXAMPLE
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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:
{{1,2,3,4,5,6}}
{{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}}
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MATHEMATICA
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A283877 = Import["https://oeis.org/A283877/b283877.txt", "Table"][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A319056
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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.
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+10
47
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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EXAMPLE
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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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A293606
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Number of unlabeled antichains of weight n.
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+10
46
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OFFSET
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0,3
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COMMENTS
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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.
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)
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LINKS
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FORMULA
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EXAMPLE
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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)).
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A302545
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Number of non-isomorphic multiset partitions of weight n with no singletons.
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+10
46
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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;
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text;
internal format)
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OFFSET
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0,3
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COMMENTS
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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}}
{{1,1},{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}}
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LINKS
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EXAMPLE
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The a(4) = 12 multiset partitions:
{{1,1,1,1}}
{{1,1,2,2}}
{{1,2,2,2}}
{{1,2,3,3}}
{{1,2,3,4}}
{{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}}
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A306021
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Number of set-systems spanning {1,...,n} in which all sets have the same size.
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+10
43
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1, 1, 2, 6, 54, 1754, 1102746, 68715913086, 1180735735356265746734, 170141183460507906731293351306656207090, 7237005577335553223087828975127304177495735363998991435497132232365910414322
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of labeled uniform hypergraphs spanning n vertices. - Andrew Howroyd, Jan 16 2024
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)).
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EXAMPLE
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The a(3) = 6 set-systems in which all sets have the same size:
{{1,2,3}}
{{1}, {2}, {3}}
{{1,2}, {1,3}}
{{1,2}, {2,3}}
{{1,3}, {2,3}}
{{1,2}, {1,3}, {2,3}}
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MATHEMATICA
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Table[Sum[(-1)^(n-k)*Binomial[n, k]*(1+Sum[2^Binomial[k, d]-1, {d, k}]), {k, 0, n}], {n, 12}]
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PROG
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(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
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CROSSREFS
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Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306019, A306020.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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