cp's OEIS Frontend

A054515 Number of ways to place non-intersecting diagonals in convex (n+2)-gon so as to create no quadrilaterals.

%I A054515 #54 Nov 26 2024 13:25:31
%S A054515 1,1,2,6,21,78,301,1198,4888,20340,85986,368239,1594183,6965380,
%T A054515 30675399,136026759,606848034,2721783023,12265670909,55511013680,
%U A054515 252193872912,1149742659556,5258257323304,24117924005616,110915268468358,511334146237807,2362650323603539
%N A054515 Number of ways to place non-intersecting diagonals in convex (n+2)-gon so as to create no quadrilaterals.
%C A054515 Number of tree interval posets of permutations with n+1 minimal elements. - _Mathilde Bouvel_, Oct 21 2021
%H A054515 Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, <a href="https://arxiv.org/abs/2303.13115">Blockwise simple permutations</a>, arXiv:2303.13115 [math.CO], 2023.
%H A054515 Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, <a href="https://arxiv.org/abs/2411.13193">Geometric view of interval poset permutations</a>, arXiv:2411.13193 [math.CO], 2024. See pp. 3, 8.
%H A054515 Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="http://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv preprint arXiv:1503.05242 [math.CO], 2015
%H A054515 Mathilde Bouvel, Lapo Cioni, and Benjamin Izart, <a href="https://arxiv.org/abs/2110.10000">The interval posets of permutations seen from the decomposition tree perspective</a>, arXiv:2110.10000 [math.CO], 2021.
%H A054515 Len Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/vsd3.html">Generalization and some variants</a>, see Quad-free.
%H A054515 Bridget Eileen Tenner, <a href="https://arxiv.org/abs/2007.06142">Interval posets for permutations</a>, arXiv:2007.06142 [math.CO], 2020-2021.
%F A054515 REVERT transform of (1-2*x+x^2-x^3)/(1-x) [Smiley].
%F A054515 a(n-1) = (1/n) * [binomial(2n-2,n-1) + Sum_{i=1..(n-3)} Sum_{k=1..Min(i,(n-i-1)/2)} binomial(n+i-1,i)*binomial(i,k)*binomial(n-i-k-2,k-1) ] if n>1. Proved in M. Bouvel, L. Cioni, B. Izart (Theorem 21) with offset 1. - _Mathilde Bouvel_, Oct 21 2021
%F A054515 G.f. A(z) = Sum_{n>=0} a(n)*z^n satisfies A(z) = 1 + z*A^2 + z^3*A^4/(1-z*A). Proved in M. Bouvel, L. Cioni, B. Izart (Equation (6) page 17 with offset 1). - _Mathilde Bouvel_, Oct 21 2021
%F A054515 Asymptotic behavior of a(n-1) is c*n^(-3/2)*r^n with c approximately 0.0792 and r approximately 4.8920. Proved in M. Bouvel, L. Cioni, B. Izart (Theorem 22). - _Mathilde Bouvel_, Oct 21 2021
%F A054515 D-finite with recurrence 23 *n *(n-1) *(12869043*n-33144451) *(n+1) *a(n) -n *(n-1) *(1989552043*n^2-6117767430*n+2643232213) * a(n-1) +(n-1) *(3359030609*n^3-15361701516*n^2+20123332181*n-6949961920) *a(n-2) +(-3560897749*n^4+25182507306*n^3-62054513365*n^2 +60006265908*n-16495478980) *a(n-3) +3*(146027817*n^4-1247820696*n^3+3378236999*n^2-2363753280*n-1468123920)*a(n-4) -3*(335627*n+695280) *(3*n-13) *(3*n-11) *(n-4) *a(n-5)=0. - _R. J. Mathar_, Oct 28 2021
%F A054515 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(2*n-k,n-3*k). - _Seiichi Manyama_, Jan 26 2024
%e A054515 a(3) = 6 because the pentagon allows null placement and five ways to place two diagonals.
%p A054515 read("transforms") :
%p A054515 taylor( (1-2*y+y^2-y^3)/(1-y),y=0,50) ;
%p A054515 gfun[seriestolist](%) ;
%p A054515 REVERT(%) ; # _R. J. Mathar_, Nov 04 2021
%t A054515 InverseSeries[Series[(y-2*y^2+y^3-y^4)/(1-y), {y, 0, 24}], x] (* then A(x)=[y(x)-x]/x *)
%o A054515 (PARI) my(N=28, x='x+O('x^N)); Vec(serreverse((x-2*x^2+x^3-x^4)/(1-x))) \\ _Hugo Pfoertner_, Jan 26 2024
%Y A054515 Cf. A046736, A049124, A003168, A054514, A348479 (free interv. posets not necess. trees).
%K A054515 nonn
%O A054515 0,3
%A A054515 _Len Smiley_, Apr 08 2000
%E A054515 a(0) = 1 prefixed by _R. J. Mathar_, Nov 04 2021