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A003450 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation and reflection.
(Formerly M1673)
6

%I M1673

%S 1,2,6,24,89,371,1478,6044,24302,98000,392528,1570490,6264309,

%T 24954223,99253318,394409402,1565986466,6214173156,24647935156,

%U 97732340680,387428854374,1535588541762,6085702368796,24116801236744,95569050564444,378718095630676

%N Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation and reflection.

%C In other words, the number of (n - 5)-dissections of an n-gon modulo the dihedral action.

%C Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the dihedral action.

%C The dissection will always be composed of either 1 pentagon and n-5 triangles or 2 quadrilaterals and n-6 triangles. - _Andrew Howroyd_, Nov 24 2017

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A003450/b003450.txt">Table of n, a(n) for n = 5..200</a>

%H D. Bowman and A. Regev, <a href="http://arxiv.org/abs/1209.6270">Counting symmetry classes of dissections of a convex regular polygon</a>, arXiv:1209.6270, 2012.

%H P. Lisonek, <a href="http://dx.doi.org/10.1006/jsco.1995.1066">Closed forms for the number of polygon dissections</a>, Journal of Symbolic Computation 20 (1995), 595-601.

%H C. R. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. math. 18 (1978) 370-388.

%F See Maple program.

%p C:=n->binomial(2*n,n)/(n+1);

%p T32:=proc(n) local t1; global C;

%p if n mod 2 = 0 then

%p t1 := (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));

%p if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;

%p if n mod 2 = 0 then t1:=t1+((3*(n-4)*(n-1))/(16*(n-3)))*C(n/2-1) fi;

%p else

%p t1 := (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));

%p if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;

%p if n mod 2 = 1 then t1:=t1+((n^2-2*n-11)/(8*(n-4)))*C((n-3)/2) fi;

%p fi;

%p t1; end;

%p [seq(T32(n),n=5..40)];

%t c = CatalanNumber;

%t T32[n_] := Module[{t1}, If[EvenQ[n], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5)*c[n/5-1]]; If[EvenQ[n], t1 = t1 + ((3*(n-4)*(n-1))/(16*(n-3)))*c[n/2-1]], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n *(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5) * c[n/5-1]]; If[OddQ[n], t1 = t1 + ((n^2 - 2*n - 11)/(8*(n-4)))*c[(n-3)/2]]]; t1];

%t Table[T32[n], {n, 5, 40}] (* _Jean-Fran├žois Alcover_, Dec 11 2017, translated from Maple *)

%o (PARI) \\ See A295419 for DissectionsModDihedral()

%o { my(v=DissectionsModDihedral(apply(i->if(i>=3&&i<=5, y^(i-3) + O(y^3)), [1..30]))); apply(p->polcoeff(p, 2), v[5..#v]) } \\ _Andrew Howroyd_, Nov 24 2017

%Y A diagonal of A295634.

%Y Cf. A003449, A295419.

%Y Cf. A003444, A003445, A220881.

%K nonn

%O 5,2

%A _N. J. A. Sloane_

%E Entry revised (following Bowman and Regev) by _N. J. A. Sloane_, Dec 28 2012

%E Name clarified by _Andrew Howroyd_, Nov 24 2017

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Last modified April 26 12:00 EDT 2019. Contains 322472 sequences. (Running on oeis4.)