login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A268814 Number of purely crossing partitions of [n]. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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)

LINKS

Table of n, a(n) for n=0..27.

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

FORMULA

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)

EXAMPLE

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 +...

MATHEMATICA

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-Fran├žois 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 *)

PROG

(PARI) 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); }

(PARI) {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

(PARI) {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

CROSSREFS

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

Sequence in context: A279511 A281698 A333895 * A165517 A197788 A197661

Adjacent sequences:  A268811 A268812 A268813 * A268815 A268816 A268817

KEYWORD

nonn

AUTHOR

Michel Marcus, Feb 14 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 11 12:47 EDT 2021. Contains 343791 sequences. (Running on oeis4.)