A268815 - OEIS

A268815

Number of purely crossing + partitions of [n].

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

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

OFFSET

0,7

COMMENTS

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)

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 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)

EXAMPLE

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 +...

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]; 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 *)

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)); Vec(b); }

(PARI) {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

(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)*A +x*O(x^n))^m) ); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 07 2016

CROSSREFS