A268814 -id:A268814 - cp's OEIS frontend

A099947 Number of topologically connected set partitions of {1,...,n}.

1, 1, 1, 1, 2, 6, 21, 85, 385, 1907, 10205, 58455, 355884, 2290536, 15518391, 110283179, 819675482, 6355429550, 51293023347, 430062712439, 3739408304962, 33665192703946, 313354708842791, 3011545611755271, 29847401178719637, 304713973031878687, 3201007359886598431

Offset: 0

N. J. A. Sloane, Nov 12 2004

A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y. - Gus Wiseman, Feb 19 2019

			O.g.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 21*x^6 + 85*x^7 +...
From _Paul D. Hanna_, Apr 16 2013: (Start)
The o.g.f. satisfies
(1) A(x) = 1 + x/A(x) + 2*x^2/A(x)^2 + 5*x^3/A(x)^3 + 15*x^4/A(x)^4 + 52*x^5/A(x)^5 + 203*x^6/A(x)^6 + ... + A000110(n)*x^n/A(x)^n + ...
(2) A(x) = 1 + x/(A(x)-x) + x^2/((A(x)-x)*(A(x)-2*x)) + x^3/((A(x)-x)*(A(x)-2*x)*(A(x)-3*x)) + x^4/((A(x)-x)*(A(x)-2*x)*(A(x)-3*x)*(A(x)-4*x)) + ... (End)
From _Gus Wiseman_, Feb 19 2019: (Start)
The a(1) = 1 through a(6) = 21 topologically connected set partitions:
  {{1}}  {{12}}  {{123}}  {{1234}}    {{12345}}    {{123456}}
                          {{13}{24}}  {{124}{35}}  {{1235}{46}}
                                      {{13}{245}}  {{124}{356}}
                                      {{134}{25}}  {{1245}{36}}
                                      {{135}{24}}  {{1246}{35}}
                                      {{14}{235}}  {{125}{346}}
                                                   {{13}{2456}}
                                                   {{134}{256}}
                                                   {{1345}{26}}
                                                   {{1346}{25}}
                                                   {{135}{246}}
                                                   {{1356}{24}}
                                                   {{136}{245}}
                                                   {{14}{2356}}
                                                   {{145}{236}}
                                                   {{146}{235}}
                                                   {{15}{2346}}
                                                   {{13}{25}{46}}
                                                   {{14}{25}{36}}
                                                   {{14}{26}{35}}
                                                   {{15}{24}{36}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..222
Janet Simpson Beissinger, The enumeration of irreducible combinatorial objects, J. Comb. Theory, Ser. A, 38 (1985), pp. 143-169. (Example 6.2)
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016. See column NIS in Table 2 p. 8.
FindStat, The number of topologically connected components of a set partition.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Martin Klazar, Bell numbers, their relatives and algebraic differential equations
Martin Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.

Cf. A000108, A000110, A001187, A007297, A016098, A092918, A268814, A268815, A305078, A306438, A323818, A324166, A324172, A324173.

a[0] = 1; a[n_] := Module[{A = 1 + x}, For[i = 1, i <= n, i++, A = Sum[x^m/Product[A - k*x + x*O[x]^n, {k, 1, m}], {m, 0, n}]]; Coefficient[A, x^n]]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 13 2013, after Paul D. Hanna *)
nn=8;
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Solve[Table[BellB[n]==Sum[Product[a[Length[s]],{s,stn}],{stn,Select[sps[Range[n]],nonXQ]}],{n,nn}],Array[a,nn]] (* Gus Wiseman, Feb 19 2019 *)

{a(n)=if(n<0, 0, polcoeff( x/serreverse(x*serlaplace(exp(exp(x+x*O(x^n))-1))), n))} /* Michael Somos, Sep 22 2005 */

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/prod(k=1, m, A - k*x +x*O(x^n)) )); polcoeff(A, n)} \\ Paul D. Hanna, Apr 16 2013

From Paul D. Hanna, Apr 16 2013: (Start)
O.g.f. A(x) satisfies
(1) A(x) = Sum_{n>=0} A000110(n)*x^n/A(x)^n, where A000110 are the Bell numbers.
(2) A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (A(x) - k*x).
(3) A(x) = 1/(1 - x/(A(x) - 1*x/(1 - x/(A(x) - 2*x/(1 - x/(A(x) - 3*x/(1 - x/(A(x) - 4*x/(1 - x/(A(x) - ... )))))))))), a continued fraction. (End)
B(n) = Sum_p Product_{s in p} a(|s|) where p is a non-crossing set partition of {1,...,n} and B = A000110. In words, every set partition of {1,...,n} can be uniquely decomposed as a non-crossing set partition together with a topologically connected set partition of each block. - Gus Wiseman, Feb 19 2019

Name edited by Gus Wiseman, Feb 19 2019

A324327 Number of topologically connected chord graphs covering {1,...,n}.

Original entry on oeis.org

1, 0, 1, 0, 1, 11, 257

Offset: 0

Gus Wiseman, Feb 22 2019

A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.

Covering means there are no isolated vertices.

			The a(0) = 1 through a(5) = 11 graphs:
  {}  {{12}}  {{13}{24}}  {{13}{14}{25}}
                          {{13}{24}{25}}
                          {{13}{24}{35}}
                          {{14}{24}{35}}
                          {{14}{25}{35}}
                          {{13}{14}{24}{25}}
                          {{13}{14}{24}{35}}
                          {{13}{14}{25}{35}}
                          {{13}{24}{25}{35}}
                          {{14}{24}{25}{35}}
                          {{13}{14}{24}{25}{35}}

Gus Wiseman, The a(5) = 11 topologically connected chord graphs.
Gus Wiseman, The a(6) = 257 topologically connected chord graphs.

Cf. A000108, A000699 (the case with disjoint edges), A001764, A002061, A007297, A016098, A054726, A099947, A136653 (the case with set-theoretical connectedness also), A268814.

Cf. A324167, A324169 (non-crossing covers), A324172, A324173, A324323, A324328 (non-covering case).

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[crosscmpts[#]]<=1]&]],{n,0,5}]

Inverse binomial transform of A324328.

A324328 Number of topologically connected chord graphs on a subset of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 4, 8, 27, 354

Offset: 0

Gus Wiseman, Feb 22 2019

			The a(0) = 1 through a(5) = 27 graphs:
  {}  {}  {}      {}      {}          {}
          {{12}}  {{12}}  {{12}}      {{12}}
                  {{13}}  {{13}}      {{13}}
                  {{23}}  {{14}}      {{14}}
                          {{23}}      {{15}}
                          {{24}}      {{23}}
                          {{34}}      {{24}}
                          {{13}{24}}  {{25}}
                                      {{34}}
                                      {{35}}
                                      {{45}}
                                      {{13}{24}}
                                      {{13}{25}}
                                      {{14}{25}}
                                      {{14}{35}}
                                      {{24}{35}}
                                      {{13}{14}{25}}
                                      {{13}{24}{25}}
                                      {{13}{24}{35}}
                                      {{14}{24}{35}}
                                      {{14}{25}{35}}
                                      {{13}{14}{24}{25}}
                                      {{13}{14}{24}{35}}
                                      {{13}{14}{25}{35}}
                                      {{13}{24}{25}{35}}
                                      {{14}{24}{25}{35}}
                                      {{13}{14}{24}{25}{35}}

Cf. A000108, A000699, A001764, A002061, A007297, A016098, A054726 (non-crossing chord graphs), A099947, A136653, A268814.

Cf. A324168, A324169, A324172, A324173, A324323, A324327 (covering case).

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[crosscmpts[#]]<=1&]],{n,0,5}]

Binomial transform of A324327.

A324323 Regular triangle read by rows where T(n,k) is the number of topologically connected set partitions of {1,...,n} with k blocks, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 5, 0, 0, 0, 0, 1, 16, 4, 0, 0, 0, 0, 1, 42, 42, 0, 0, 0, 0, 0, 1, 99, 258, 27, 0, 0, 0, 0, 0, 1, 219, 1222, 465, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Feb 22 2019

A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y.

			Triangle begins:
    1
    0    1
    0    1    0
    0    1    0    0
    0    1    1    0    0
    0    1    5    0    0    0
    0    1   16    4    0    0    0
    0    1   42   42    0    0    0    0
    0    1   99  258   27    0    0    0    0
    0    1  219 1222  465    0    0    0    0    0
Row n = 6 counts the following set partitions:
  {{123456}}  {{1235}{46}}  {{13}{25}{46}}
              {{124}{356}}  {{14}{25}{36}}
              {{1245}{36}}  {{14}{26}{35}}
              {{1246}{35}}  {{15}{24}{36}}
              {{125}{346}}
              {{13}{2456}}
              {{134}{256}}
              {{1345}{26}}
              {{1346}{25}}
              {{135}{246}}
              {{1356}{24}}
              {{136}{245}}
              {{14}{2356}}
              {{145}{236}}
              {{146}{235}}
              {{15}{2346}}

Row sums are A099947. Row k = 2 is A002662.
Cf. A000108, A000110, A001263, A016098, A136653, A268814, A268815, A306438, A324011.
Cf. A324166, A324172, A324173, A324327, A324328.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[crosscmpts[#]]<=1&&Length[#]==k&]],{n,0,6},{k,0,n}]

A268815 Number of purely crossing + partitions of [n].

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 5, 19, 76, 360, 1792, 9634, 55286, 336396, 2162554, 14629720, 103818489, 770678553, 5969822993, 48148947503, 403545713463, 3508356996105, 31587389832791, 294087418038113, 2827471212909189, 28037001032306431, 286398141349873925, 3010540174760962975

Offset: 0

Michel Marcus, Feb 14 2016

nonn

For the definition of these special purely crossing partitions refer to Dykema link (see PC+(n) Definition 2.1 and Table 2).
From Gus Wiseman, Feb 23 2019: (Start)
a(n) is the number of topologically connected (A099947) set partitions of {1,...,n} with no successive elements in the same block. For example, the a(4) = 1 through a(7) = 19 set partitions are:
  {{13}{24}}  {{135}{24}}  {{135}{246}}    {{1357}{246}}
                           {{13}{25}{46}}  {{13}{246}{57}}
                           {{14}{25}{36}}  {{13}{257}{46}}
                           {{14}{26}{35}}  {{135}{26}{47}}
                           {{15}{24}{36}}  {{135}{27}{46}}
                                           {{136}{24}{57}}
                                           {{136}{25}{47}}
                                           {{137}{25}{46}}
                                           {{14}{257}{36}}
                                           {{14}{26}{357}}
                                           {{146}{25}{37}}
                                           {{146}{27}{35}}
                                           {{147}{25}{36}}
                                           {{147}{26}{35}}
                                           {{15}{246}{37}}
                                           {{15}{247}{36}}
                                           {{157}{24}{36}}
                                           {{16}{24}{357}}
                                           {{16}{247}{35}}
(End)

			G.f.: A(x) = 1 + x + x^4 + x^5 + 5*x^6 + 19*x^7 + 76*x^8 + 360*x^9 + 1792*x^10 +...

Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016.

Cf. A000108, A000110, A005493, A016098, A099947, A268814, A306417, A324011, A324166, A324173, A324324, A324327.

n = 30; F = x*Sum[BellB[k] x^k, {k, 0, n}] + O[x]^n; B = ComposeSeries[1/( InverseSeries[F, w] /w)-1, x/(1+x) + O[x]^n]; CoefficientList[B, x] // Rest (* Jean-François Alcover, Feb 16 2016, adapted from K. J. Dykema's code *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
intvQ[set_]:=Or[set=={},Sort[set]==Range[Min@@set,Max@@set]];
Table[Length[Select[sps[Range[n]],And[!MatchQ[#,{_,{_,x_,y_,_},_}/;x+1==y],#=={}||And@@Not/@intvQ/@Union@@@Subsets[#,{1,Length[#]-1}]]&]],{n,0,10}] (* Gus Wiseman, Feb 23 2019 *)

lista(nn) = {c = x/serreverse(x*serlaplace(exp(exp(x+x*O(x^nn)) -1))); b = subst(c, x, x/(1+x) + O(x^nn)); Vec(b);}

{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/prod(k=1, m, (1+x)*A - k*x +x*O(x^n)) )); polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

{Stirling2(n, k) = n!*polcoeff(((exp(x+x*O(x^n)) - 1)^k)/k!, n)}
{Bell(n) = sum(k=0,n, Stirling2(n, k) )}
{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, Bell(m)*x^m/((1+x)*A +x*O(x^n))^m) ); polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

G.f.: G(x) satisfies C(x) = G(x/1-x) where C(x) is the g.f. of A099947 (see B(x) in Dykema link p. 7).
From Paul D. Hanna, Mar 07 2016: (Start)
O.g.f. A(x) satisfies
(1) A(x) = Sum_{n>=0} A000110(n)*x^n/((1+x)^n*A(x)^n), where A000110 are the Bell numbers.
(2) A(x) = Sum_{n>=0} x^n / Product_{k=1..n} ((1+x)*A(x) - k*x).
(3) A(x) = 1/(1 - x/((1+x)*A(x) - 1*x/(1 - x/((1+x)*A(x) - 2*x/(1 - x/((1+x)*A(x) - 3*x/(1 - x/((1+x)*A(x) - 4*x/(1 - x/((1+x)*A(x) - ... )))))))))), a continued fraction. (End)

cp's OEIS Frontend

A099947 Number of topologically connected set partitions of {1,...,n}.

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A324327 Number of topologically connected chord graphs covering {1,...,n}.

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A324328 Number of topologically connected chord graphs on a subset of {1,...,n}.

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A324323 Regular triangle read by rows where T(n,k) is the number of topologically connected set partitions of {1,...,n} with k blocks, 0 <= k <= n.

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A268815 Number of purely crossing + partitions of [n].

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