A324324 -id:A324324 - cp's OEIS frontend

A268814 Number of purely crossing partitions of [n].

1, 0, 0, 0, 1, 0, 5, 14, 62, 298, 1494, 8140, 47146, 289250, 1873304, 12756416, 91062073, 679616480, 5290206513, 42858740990, 360686972473, 3147670023632, 28439719809159, 265647698228954, 2561823514680235, 25475177517626196, 260922963832247729, 2749617210928715246

Offset: 0

Michel Marcus, Feb 14 2016

nonn

For the definition of a purely crossing partition refer to Dykema link (see PC(n) Definition 1.2 and Table 2).
From Gus Wiseman, Feb 23 2019: (Start)
For n >= 1, a set partition of {1,...,n} is purely crossing if it is topologically connected (A099947), has no successive elements in the same block (A000110(n - 1)), and the first and last vertices belong to different blocks (A005493(n - 2)). For example, the a(4) = 1, a(6) = 5, and a(7) = 14 purely crossing set partitions are:
  {{13}{24}}  {{135}{246}}    {{13}{246}{57}}
              {{13}{25}{46}}  {{13}{257}{46}}
              {{14}{25}{36}}  {{135}{26}{47}}
              {{14}{26}{35}}  {{135}{27}{46}}
              {{15}{24}{36}}  {{136}{24}{57}}
                              {{136}{25}{47}}
                              {{14}{257}{36}}
                              {{14}{26}{357}}
                              {{146}{25}{37}}
                              {{146}{27}{35}}
                              {{15}{246}{37}}
                              {{15}{247}{36}}
                              {{16}{24}{357}}
                              {{16}{247}{35}}
(End)

			G.f.: A(x) = 1 + x^4 + 5*x^6 + 14*x^7 + 62*x^8 + 298*x^9 + 1494*x^10 + 8140*x^11 + 47146*x^12 +...

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

Cf. A000108 (non-crossing partitions), A000110, A000699, A001263, A002662, A005493, A016098, A054726, A099947, A268815, A306417, A324011, A324166, A324172, A324173, A324324.

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]; A = (B-x)/(1+x); Join[{1}, CoefficientList[A, x] // Rest] (* Jean-François Alcover, Feb 23 2016, adapted from K. J. Dykema's code *)
intvQ[set_]:=Or[set=={},Sort[set]==Range[Min@@set,Max@@set]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],And[!MatchQ[#,{_,{_,x_,y_,_},_}/;x+1==y],#=={}||And@@Not/@intvQ/@Union@@@Subsets[#,{1,Length[#]-1}],#=={}||Position[#,1][[1,1]]!=Position[#,n][[1,1]]]&]],{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)); vb = Vec(b-1); va = vector(#vb); va[1] = 0; va[2] = 0; for (k=3, #va, va[k] = vb[k] - va[k-1]; ); concat(1, va); }

{a(n) = my(A=1+x^3); for(i=1, n, A = sum(m=0, n, x^m/prod(k=1, m, (1+x)^2*A - k*x +x*O(x^n)) )/(1+x) ); polcoeff( A, n)}
for(n=0,35,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 +x*O(x^n))^(2*m+1)*A^m)) ); polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

G.f.: G(x) satisfies B(x) = x + (1 + x)*G(x) where B(x) is the g.f. of A268815 (see A(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)^(2*n+1) * A(x)^n), where A000110 are the Bell numbers.
(2) A(x) = 1/(1+x) * Sum_{n>=0} x^n / Product_{k=1..n} ((1+x)^2*A(x) - k*x).
(3) A(x) = 1/(1+x - x/((1+x)*A(x) - 1*x/(1+x - x/((1+x)*A(x) - 2*x/(1+x - x/((1+x)*A(x) - 3*x/(1+x - x/((1+x)*A(x) - 4*x/(1+x - x/((1+x)*A(x) -...)))))))))), a continued fraction. (End)

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)

A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 36, 0, 14, 0, 0, 0, 25, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 103, 0, 0, 0, 0, 0, 0, 0, 75

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A multiset partition is crossing if it contains two blocks of the form {{...x...y...},{...z...t...}} with x < z < y < t or z < x < t < y.

			The a(36) = 10 crossing multiset partitions of {1,1,2,2,3,4}:
  {{1,3},{1,2,2,4}}
  {{2,4},{1,1,2,3}}
  {{1,1,3},{2,2,4}}
  {{1,2,3},{1,2,4}}
  {{1},{1,3},{2,2,4}}
  {{1},{2,4},{1,2,3}}
  {{2},{1,3},{1,2,4}}
  {{2},{1,1,3},{2,4}}
  {{1,2},{1,3},{2,4}}
  {{1},{2},{1,3},{2,4}}

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324325.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

a(n) + A324325(n) = A318284(n).

A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 14, 15, 26, 22, 21, 29, 19, 30, 33, 31, 30, 66, 38, 42, 52, 56, 42, 47, 45, 57, 82, 77, 67, 77, 67, 101, 98, 135, 64, 137, 97, 176, 104, 109, 109, 118, 105, 231, 213, 97, 127, 181, 139, 297, 173, 385, 195, 269

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

A multiset partition is crossing if it contains two blocks of the form {{...x...y...},{...z...t...}} where x < z < y < t or z < x < t < y.

			The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,4},{2,3}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{1,2},{4}}
  {{1},{2},{3},{4}}
Missing from this list is {{1,3},{2,4}}.

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324326.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

a(n) + A324326(n) = A318284(n).

A306551 Number of non-double-crossing set partitions of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 863, 3999, 19880, 105134, 587479, 3449505

Offset: 0

Gus Wiseman, Feb 23 2019

Two blocks of a set partitions double-cross each other if they are of the form {{...a...b...c...},{...x...y...z...}} for some a < x < b < y < c < z or x < a < y < b < z < c.

			Most small set partitions are not double-crossing. The smallest that is double-crossing is {{1,3,5},{2,4,6}}.

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

Cf. A000108, A000110, A001263, A002061, A005493, A007297, A016098, A054726, A099947, A324166, A324167, A324172, A324324.

nonXXQ[stn_]:=!MatchQ[stn,{_,{_,a_,_,b_,_,c_,_},_,{_,x_,_,y_,_,z_,_},_}/;a_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],nonXXQ]],{n,0,8}]

A306558 Number of double-crossing set partitions of {1,...,n}.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 14, 141, 1267, 10841, 91091, 764092

Offset: 0

Gus Wiseman, Feb 23 2019

Two blocks of a set partitions double-cross each other if they are of the form {{...a...b...c...},{...x...y...z...}} for some a < x < b < y < c < z or x < a < y < b < z < c.

			The a(7) = 14 double-crossing set partitions:
  {{1,3,5},{2,4,6,7}}
  {{1,3,6},{2,4,5,7}}
  {{1,4,6},{2,3,5,7}}
  {{1,2,4,6},{3,5,7}}
  {{1,3,4,6},{2,5,7}}
  {{1,3,5,6},{2,4,7}}
  {{1,3,5,7},{2,4,6}}
  {{1},{2,4,6},{3,5,7}}
  {{1,3,5},{2,4,6},{7}}
  {{1,3,5},{2,4,7},{6}}
  {{1,3,6},{2,4,7},{5}}
  {{1,3,6},{2,5,7},{4}}
  {{1,4,6},{2},{3,5,7}}
  {{1,4,6},{2,5,7},{3}}

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

Cf. A000108, A000110, A001263, A002061, A005493, A007297, A016098 (crossing set partitions), A054726, A099947, A324166, A324172, A324324.

croXXQ[stn_]:=MatchQ[stn,{_,{_,a_,_,b_,_,c_,_},_,{_,x_,_,y_,_,z_,_},_}/;a_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],croXXQ]],{n,0,8}]

cp's OEIS Frontend

A268814 Number of purely crossing partitions of [n].

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

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A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A306551 Number of non-double-crossing set partitions of {1,...,n}.

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A306558 Number of double-crossing set partitions of {1,...,n}.

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