OFFSET
0,3
COMMENTS
There is also a sum-formula for a(n). See Bousquet-Mélou and Xin.
Also partitions avoiding a certain pattern (see J. Bloom and S. Elizalde). - N. J. A. Sloane, Jan 02 2013
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jonathan Bloom and Sergi Elizalde, Pattern avoidance in matchings and partitions, arXiv preprint arXiv:1211.3442 [math.CO], 2012.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury, Yi Zhang, On sequences associated to the invariant theory of rank two simple Lie algebras, arXiv:1911.10288 [math.CO], 2019.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, On some combinatorial sequences associated to invariant theory, arXiv:2110.13753 [math.CO], 2021.
M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, arXiv:math/0506551 [math.CO], 2005-2006.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011.
Wei Chen, Enumeration of Set Partitions Refined by Crossing and Nesting Numbers, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014.
W. Chen, E. Deng, R. Du, R. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
Juan B. Gil, Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018.
Zhicong Lin, Restricted inversion sequences and enhanced 3-noncrossing partitions, arXiv:1706.07213 [math.CO], 2017.
Zhicong Lin, Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
T. Mansour and M. Shattuck, Some enumerative results related to ascent sequences, arXiv preprint arXiv:1207.3755 [math.CO], 2012. - N. J. A. Sloane, Dec 22 2012
Marni Mishna and Lily Yen, Set partitions with no k-nesting, arXiv preprint arXiv:1106.5036 [math.CO], 2011.
FORMULA
Recurrence: (9*n^2+27*n) * a(n) + (-10*n^2-64*n-84) * a(n+1) + (n^2+13*n+42) * a(n+2) = 0.
a(n) = (-18*(n+1)*(4*n^5+73*n^4+530*n^3+1928*n^2+3654*n+2916)*A002893(n)+(8*n^6+17156*n^2+6084*n^3+17496+27612*n+1358*n^4+162*n^5) *A002893(n+1))/ (3*n*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)). - Mark van Hoeij, Nov 05 2011
G.f.: (1+7*x-20*x^2+30*x^3-18*x^4-(3*x+1)^2*(x-1)^2*hypergeom([-2/3, -1/3],[2],27*x*(x-1)^2/(3*x+1)^3))/(6*x^4). - Mark van Hoeij, Nov 05 2011
a(n) ~ 5 * sqrt(3) * 3^(2*n+9) / (32*Pi*n^7), Bousquet-Mélou and Xin, 2006. - Vaclav Kotesovec, Aug 23 2014
EXAMPLE
There are 203 partitions of 6 elements, but a(6)=202 because the partition (1,4)(2,5)(3,6) has a 3-crossing.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 202*x^6 + 859*x^7 + ...
MATHEMATICA
a[0] = a[1] = 1; a[n_] := a[n] = (2*(5*n^2 + 12*n - 2)*a[n-1] + 9*(-n^2 + n + 2)*a[n-2])/((n+4)*(n+5)); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 30 2015 *)
PROG
(PARI)
v = vector(66, n, n);
for (n=1, #v-2, v[n+2] = ((10*n^2+64*n+84)*v[n+1]-(9*n^2+27*n)*v[n]) / (n^2+13*n+42) );
vector(#v+1, n, if(n==1, 1, v[n-1])) \\ Joerg Arndt, Sep 01 2012
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Mireille Bousquet-Mélou, Jun 29 2005
EXTENSIONS
More terms added by Joerg Arndt, Sep 01 2012
STATUS
approved