# Polya theory for orbiquotient sets.

1. IntroductionAssume that a finite group G acts on a finite set X. The quotient X/G of the action of G on X is a rich and subtle concept, traditionally X/G = {[bar.x]|x [member of] X} where [bar.x] = {gx|x [member of] X}. In recent years it has proven convenient to modify this notion in various contexts. For example one may think of X/G as the groupoid whose set of objects is X and with morphisms given by X/G(a, b) = {g [member of] G|ga = b} for a,b [member of] G. Following Connes [8] the groupoid X/G is studied with the methods of non- commutative geometry, i.e. looking at the convolution (incidence) algebra of X/G. Another approach to quotient sets became rather popular after Vafa and Witten introduced in [17] the so called stringy Euler numbers. In a nutshell they considered the Euler numbers of orbiquotient sets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where C(G) is the set of conjugacy classes of G, [X.sup.g] [subset or equal to] X is the set of points fixed by g [member of] G, and Z(g) is the centralizer of g in G. We will assume that a representative g [member of] G has been chosen for each conjugacy class [bar.g] of G. Orbiquotient sets first appeared, see [2], in the context of equivariantK-theory in the works of Atiyah and Segal. The goal of this paper is to bring the notion of orbiquotient sets into combinatorial waters. Let us provide a combinatorial motivation for the study of orbiquotient sets inspired by an analogue topological construction given by Hirzebruch and Hofer in [14]. Consider the set of n-cycles in X/G, i.e., the set of maps f: Z [right arrow] X/G such that f(k) = f(k + n) for k [member of] Z. Suppose we want to lift f to a map l: Z [right arrow] X such that [pi] * l = f, where [pi]: X [right arrow] X/G is the canonical projection. The lift l will not be unique, indeed if l is a lift then gl is another a lift; also we have that l(k) = gl(k + n) for all k [member of] Z and some g [member of] G. Thus the set of lifts of n-periodic maps Z [right arrow] X/G may be identified with

{l: Z [right arrow] X|l(k) = gl(k + n) for k [member of] Z and some g [member of] G}/G.

Inside the later set sits I(G, X)/G the set of constant maps, where

I(G,X) = {(g,x) [member of] G x X|gx = x}

is the so called inertial set [15]. The group G acts on I(G, X) as k(g, x) = (kg[k.sup.-1], kx), and it is not hard to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this paper we develop orbianalogues for two main results in elementary combinatorics, the orbit counting lemma and the Polya-Redfield theorem, see [4, 16]. We fix a commutative ring A and consider the category of A-weighted sets whose objects are pairs (X, f) where X is a finite set and f: X [right arrow] A is an arbitrary map called the weight of X. Morphisms between A-weighted sets are weight preserving bijections. The cardinality [[absolute value of X].sub.f] of a weighted set (X, f) is given by

[[absolute value of X].sub.f] = [summation over (x[member of]X)] f(x).

A finite group G acts on (X, f) if G acts on X and f(gx) = f(x) for all g [member of] G, x [member of] X. The Cauchy-Frobenius-Burnside orbit counting lemma gives us a way to compute [[absolute value of X/G].sub.f] as follows:

[[absolute value of X/G].sub.f] = [1/[absolute value of G]] [summation over (g[member of]G)] [[absolute value of [X.sup.g]].sub.f].

Suppose now that G is a group of permutations G [subset] [S.sub.m]. The cardinality of [X.sup.m]/G is determined by the Polya-Redfield theorem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [P.sub.G] is the cycle index polynomial of G given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [c.sub.i](g) is the number of g-cycles of length i. If X = [n] and f(i) = [x.sub.i] for i [member of] X, then directly from the definition of quotient sets we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [c.sub.G] ([i.sub.1],...,[i.sub.n]) counts the colorations of [m] with [i.sub.k] elements of color k [member of] [n], and two colorations are identified if they are linked by the action of G. The Polya- Redfield theorem allows us to compute the coefficients [c.sub.G] ([i.sub.1],...,[i.sub.n]) in a different way, namely we have that

[[absolute value of [[n].sup.m]/G].sub.f] = [P.sub.G] ([n.summation over (j=1)] [x.sub.j], [n.summation over (j=1)] [x.sup.2.sub.j],..., [n.summation over (j=1)] [x.sup.m.sub.j]).

The rest of this work is organized as follows. In Section 2 we provide an orbi- analogue of the orbit counting lemma. In Section 3 we provide an orbi-analogue of the Polya-Redfield theorem in full generality, we shall see that lattice of partitions plays a fundamental role in our presentation. In the remaining sections we explicitly compute the orbicycle index polynomial for various groups in increasing order of difficulty. In Section 4 we consider the case of cyclic groups, and apply it to the study of orbicycles in orbiquotient sets. In Section 5 we consider the full symmetric group. In a rather dull fashion we may regard combinatorics as geometry in dimension zero. It is thus rather interesting when one can show that the zero dimensional combinatorial case determines the higher dimensional situation. A theorem of this sort is proved at the end of Section 5 which provides a strong motivation for the study of orbiquotient sets. In Section 6 we compute the orbicycle index polynomial for the dihedral groups.

2. Orbi-analogue of the orbit counting lemma

If S [subset or equal to] G and G acts on X, then we set [X.sup.S] = {x [member of] X|gx = x for g [member of] S}. Also we let <[g.sub.1],...,[g.sub.n]> be the subgroup of G generated by {[g.sub.1],...,[g.sub.n]} [subset] G.

Definition 1. The orbiquotient of X by the action of G is the set given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The orbiquotient X[/.sup.orb] G is well defined up to canonical bijections. Indeed if h = kg[k.sup.-1] then the map [psi]: [X.sup.g] [right arrow] [X.sup.h] given by [psi](x) = kx induces a bijection

[psi]: [X.sup.g]/Z(g) [right arrow] [X.sup.h]/Z(h).

If G acts on a weighted set (X, f) then X[/.sup.orb]G is also weighted: [X.sup.g] is weighted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is weighted by f([bar.x]) = f(x) for [bar.x] [member of] [X.sup.g]/Z(g). Our next result is the orbi-analogue of the orbit counting lemma, let us first introduce a notation that will be used repeatedly

P(G) = {([bar.g],h)|[bar.g] [member of] C(G) and h [member of] Z(g}.

Theorem 2. If G acts on (X, f), then the cardinality of X[/.sup.orb]G is given by

[[absolute value of X[/.sup.orb]G].sub.f] = [1/[absolute value of G]] [summation over (([bar.g],h)[member of]P(G))] [absolute value of [bar.g]] [[absolute value of [X.sup.<g,h>]].sub.f].

The proof of this result is quite simple:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. Orbi-analogue of Polya-Redfield theorem

Let Par(X) be lattice of partitions of X. The minimal and maximal elements of Par(X) are {{x}|x [member of] X} and {X}, respectively. The joint [pi] [disjunction] [rho] of partitions [pi] and [rho] is defined by demanding that i, j [member of] X belong to a block of [pi] [disjunction] [rho] if there exists a sequence i = [a.sub.0],[a.sub.1],...,[a.sub.n] = j, such that for 0 [less than or equal to] i [less than or equal to] n - 1 either [a.sub.i] and [a.sub.i+1] belong to a block in [pi], or [a.sub.i] and [a.sub.i+1] belong a block in [rho]. The meet of partitions [pi] and [rho] is [pi] [disjunction] [rho] = {B [intersection] C|B [member of] [pi], C [member of] [rho], B [intersection] C [not equal to] 0}. Let the group G act on a set X with n-elements. Each g [member of] G induces a partition C(g) on X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [C.sub.i](g) = {g-cycles on X of length i} for 1 [less than or equal to] i [less than or equal to] n. We use the notation c(g) = [absolute value of C(g)] and [c.sub.i](g) = [absolute value of [C.sub.i](g)] for 1 [less than or equal to] i [less than or equal to] n. If [pi] is a partition of X we let [b.sub.k]([pi]) be the number of blocks of [pi] of cardinality k.

Definition 3. The orbicycle index polynomial [P.sup.orb.sub.G] ([x.sub.1], [x.sub.2],...) [member of] Q[[x.sub.1],[x.sub.2],...] is given by

[P.sup.orb.sub.G] ([x.sub.1], [x.sub.2],...) = [1/[absolute value of G]] [summation over (([bar.g],h)[member of]P(G))] [absolute value of [bar.g]] [x.sup.C(g)[disjunction]C(h)],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If G [subset or equal to] [S.sub.m] then G acts on [X.sup.m]. Suppose that g,h [member of] G commute, then i,j [member of] [m] belong to the same block of C(g) [disjunction] C(h) if and only if there exist a,b [member of] Z such that j = ([g.sup.a][h.sup.b])(i). It is easy to check that f [member of] [X.sup.m] is fixed by g and h if and only if f is constant on each block of C(g) [disjunction] C(h).

Theorem 4. Let (X, f) be an A-weighted set and G [subset or equal to] [S.sub.m]. The cardinality of [X.sup.m][/.sup.orb]G is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let X = [n] and f(i) = [x.sub.i], then one can check directly from the definition that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [c.sup.orb.sub.G] ([i.sub.1],...,[i.sub.n]) counts colorations c of [m] with colors in [n] such that:

* There are [i.sub.k] elements in [m] of color k [member of] [n].

* c is g-invariant for some [bar.g] [member of] C(G).

* Two g-invariant colorations are identified if they can be linked by the action of Z(g).

The orbi-analogue of the Polya-Redfield gives us another way to compute the coefficients [c.sup.orb.sub.G] ([i.sub.1],...,[i.sub.n]), namely we have that

[[absolute value of [[n].sup.m][/.sup.orb]G].sub.f] = [P.sup.orb.sub.G] ([n.summation over (j=1)] [x.sub.j], [n.summation over (j=1)] [x.sup.2.sub.j],..., [n.summation over (j=1)] [x.sup.h.sub.j]).

4. Orbicycle index polynomial of [Z.sub.n]

Let [N.sub.+] be the set of positive integers and let ([x.sub.1],[x.sub.2],...,[x.sub.k]) be the greatest common divisor of [x.sub.1],[x.sub.2],...,[x.sub.k] [member of] [N.sup.+]. The cyclic group with n-elements is denoted by [Z.sub.n]. For n, k [member of] [N.sub.+] we define an equivalence relation on [Z.sup.k.sub.n] as follows: x and y are equivalent if and only if (x, n) = (y, n). It is easy to verify that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and thus we have

[n.sup.k] = [summation over (d|n)] [absolute value of {x [member of] [Z.sup.k.sub.n]|(x,n) = d}].

For n [member of] [N.sub.+] the Jordan totient function [J.sub.k], see [1], is given by [J.sub.k](n) = [absolute value of {x [member of] [Z.sup.k.sub.n]|(x,n) = 1}]. For each d|n we have [J.sub.k](n/d) = [absolute value of {x [member of] [Z.sup.k.sub.n]|(x, n) = d}], therefore we get that [n.sup.k] = [summation over (d|n)] [J.sub.k](d). By the Mobius inversion formula [J.sub.k](n) = [summation over (d|n)][mu](n/d)[d.sup.k], thus [J.sub.k]([p.sup.r]) = [p.sup.kr]-[p.sup.k(r-1)], for p prime, and for arbitrary integer n = [p.sub.1]...[p.sub.r] we get

[J.sub.k](n) = [n.sup.k](1 - [1/[p.sup.k.sub.1]]) ... (1 - [1/[p.sup.k.sub.r]]).

We shall need the following property, an easy consequence of the previous considerations, of the Jordan totient function:

[summation over (x[member of][Z.sup.k.sub.n])] f((x,n)) = [summation over (d|n)] [J.sub.k](d)f(d),

for any f: {d: d|n} [right arrow] A. Recall that if x, y [member of] [Z.sub.n] [subset or equal to] [S.sub.n], then [absolute value of C(x) [disjunction] C(y)] = (x, y, n), and all blocks in C(x) [disjunction] C(y) are of cardinality [n/(n,x,y)]. Indeed if x [member of] [Z.sub.n], then [Z.sub.n]/(x) [congruent to] [Z.sub.(n,x)], since for a,b [member of] [Z.sub.n] we have that a = b in [Z.sub.n]/(x) if and only if a = b mod (n,x). Thus there are (n,x) blocks in [Z.sub.n]/(x) all of them of cardinality [n/(n,x)]. Similarly if x,y [member of] [Z.sub.n], then a,b [member of] [Z.sub.n] are in the same block of C(x) [disjunction] C(y) if and only if there exist r,s [member of] Z such that b = a + rx + sy mod n, or equivalently a = b mod (n, x, y).

Theorem 5.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The coefficients [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([i.sub.1],...,[i.sub.m]) are computed in [13]. As a corollary of the previous result we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let p be a prime number and r [member of] [N.sup.+], necklaces without a clasp with p beads and r colors may be identified with the set [C.sub.p][r] = [[r].sup.p]/[Z.sub.p]. As explained in [18] we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 6. The set [C.sup.orb.sub.n](X) of orbi n-cycles in X is given by [C.sup.orb.sub.n](X) = [X.sup.n][/.sup.orb][Z.sub.n].

In analogy with the example above we define the set of orbi-necklaces without a clasp with p beads and r colors to be [C.sup.orb.sub.p][r] = [[r].sup.p][/.sup.orb][Z.sub.p]. Its cardinality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next couple of results count explicitly the number of orbicycles in orbiquotient sets.

Theorem 7. If G acts on (X, f) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 8. Let (X, f) be an A-weighted set and G [subset or equal to] [S.sub.m], then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using similar methods one can count cycles on orbiquotient sets:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

5. Orbicyle index polynomial of [S.sub.n]

A partition of depth k, denoted by [alpha] [[??].sub.k] n, of n [member of] [N.sub.+] is a map [alpha]: [([N.sub.+]).sup.k] [right arrow] N such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A partition of depth 1 is a partition in the usual sense. To each partition [alpha] we associate a canonical permutation of [n] whose cycle structure is determined by [alpha]. Keeping this correspondence in mind one can check that if [alpha] [??] n then

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. If h [member of] Z([alpha]), then [b.sub.k](C([alpha]) [disjunction] C(h)) = [summation over (d|k)] [c.sub.k/d] ([[pi].sub.d](h)) where [[pi].sub.d](h) is the projection of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 9.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. By the previous remarks we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Above we used the fact that [summation over (d|k)] [c.sub.k/d] ([[pi].sub.d](h)) depends only on the cycle structure of [[pi].sub.d](h).

The orbicycle index polynomial can be use to compute the even dimensions of the orbifold cohomology groups for global orbifolds of the form [M.sup.n][/.sup.orb]G, where M is a compact smooth manifold, and G [subset] [S.sub.n]. For simplicity we only consider cohomology in even dimensions. The orbifold cohomology is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following result is a direct consequence of the characterization of the centralizer of permutations previously discussed.

Lemma 10.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assume that we are given a finite basis X for H(M), then we have the following result.

Theorem 11.

dim ([H.sup.orb] ([M.sup.n]/[S.sub.n])) = [P.sup.orb.sub.G]([absolute value of X],..., [absolute value of X]).

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice that above we use the trivial weight on X; using a generic weight we obtain further information on the orbifold cohomology groups. Theorem 12 gives a combinatorial interpretation for the orbifold cohomology groups, however we do not have a combinatorial interpretation for the orbifold product introduced by Chen and Ruan in [9]. Until recently this problem seemed hopeless, however the alternative description of the Chen-Ruan product introduced by Jarvis, Kauffman and Kimura in [15] could pave the way for such a combinatorial understanding. Theorem 11 suggests the possibility of constructing, along the lines of [11], an orbi-analogue for the symmetric functions. This issue deserves further research.

6. Orbicycle index polynomial of [D.sub.n]

The generators [rho] and [tau] of the dihedral group [D.sub.n] = {e,[rho],...,[[rho].sup.n- 1],[tau],...,[tau][[rho].sup.n-1]} are such that [[rho].sup.n] = e, [[tau].sup.2] = e and [tau][rho] = [[rho].sup.n- 1][tau]. The conjugacy classes of the dihedral groups are described in the following tables, see [10]. For n odd there are n+3/2 conjugacy classes organized in three families

Conjugacy class Representative Centralizer subgroup {e} e [D.sub.n] {[[rho].sup.i], [[rho].sup.i] {[[rho].sup.i| o [[rho].sup.-i] for 1 [less than or equal to] [less thann or i < n} equal to] i [less than or equal to] n - 1/2 {[[rho].sup.i][tau]| 0 [tau] {e, [tau]} [less than or equal to] i < n}

For n even there are n+6/2 conjugacy classes organized in five families

Conjugacy class Representative Centralizer subgroup {e} e [D.sub.n] {[[rho].sup.n/2]} [[rho].sup.n/2] [D.sub.n] {[[rho].sup.i], [[rho].sup.i] {[[rho].sup.i] | 0 [[rho].sup.-i]}, [less than or equal to] 1 [less than or i < n} equal to] i < n/2 {[[rho].sup.2i] [tau] {e, [tau], [tau]| 0 [[rho].sup.n/2], [less than or [[rho].sup.n/2][tau]} equal to] i < n/2} {[[rho].sup.2i+1] [rho][tau] {e, [rho][tau], [tau]| 0 [[rho].sup.n/2], [less than or [[rho].sup.n/2+1][tau]} equal to] i < n/2}

For [a.sub.1],...,[a.sub.k] maps from a subset of [N.sup.+] into [N.sup.+], and n in this subset we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proceeding as in the previous sections one can show that

[phi]([a.sub.1](n),...,[a.sub.k](n), n) = [summation over (d|n)][a.sub.1](d)...[a.sub.k](d)[mu] (n/d),

also for f: {d: d|n} [right arrow] A an arbitrary map we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We will use two instances of the identity above, namely, let n [member of] [N.sup.+] be odd, [a.sub.1](n) = [n-1/2], and [a.sub.2](n) = n-1, then

[phi]([a.sub.1](n), [a.sub.2](n), n) = [summation over (d|n)]([d.sup.2]-2d+1/2) [mu](n/d) = [[J.sub.2](n) - 2[phi](n)/2].

Let n [member of] [N.sup.+] be even, [a.sub.1](n) = [n-2/2], and [a.sub.2](n) = n - 1, then we get

[phi]([a.sub.1](n), [a.sub.2](n), n) = [J.sub.2](n) - 3[phi](n)/2.

Theorem 12. Let n [member of] [N.sub.+] be odd. The orbicycle index polynomial of [D.sub.n] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Recall that [tau] and [rho] are given [tau](x) = 3-x and [[rho].sup.r](x) = x + r. Our next table gives the equivalence class of x [member of] [Z.sub.n] under five different equivalence relations.

Partition Equivalence class C([tau]) [disjunction] {x,3 - x,x + [n/2], 3 - x + [n/2]} C([[rho].sup.n/2]) C([tau]) [disjunction] {x, 3 - x, 3 - x + [n/2], x + [n/2]} C([[rho].sup.n/2][tau]) C([rho][tau]) [disjunction] {x, 4 - x, x + [n/2], 4 - x + [n/2]} C([[rho].sup.n/2]) C([rho][tau]) [disjunction] {x, 4 - x, 4 - x + [n/2], x + [n/2]} C([[rho].sup.[n/2]+1][tau]) C([[rho].sup.n/2]) {x, x + [n/2], 3 - i - x, 3 - i [disjunction] - x + [n/2]} C([tau][[rho].sup.i])

So we see that the equivalence class of x [member of] [n] under the equivalence relation C([[rho].sup.n/2]) [disjunction] C([tau][[rho].sup.i]) is

[bar.x] = {x, x + [n/2], 3 - i - x, 3 - i - x + [n/2]},

so that [absolute value of [bar.x]] [member of] {2, 4}. It is not difficult to see that [absolute value of [bar.x]] = 2 if and only if either 2x [equivalent to] 3 - i or 2x [equivalent to] 3 - i + [n/2]. Therefore [b.sub.2](C([[rho].sup.n/2]) [disjunction] C([tau][[rho].sup.i])) is 1 if n/2 is odd, 2 if n/2 is even and i is odd, and 0 if n/2 is even and i is even.

Theorem 13. Let n [member of] [N.sub.+] be even. According to whether n is 0 or 2 mod 4, the orbicycle index polynomial of [D.sub.n] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We compute the last four summands in the expression above

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this work we have extended Polya theory to the context of orbiquotient sets. The main ingredient of the new theory is the orbicycle index polynomial which we computed in various cases. We expect that our construction will find applications in the study of the topology of orbifolds and also in the theory of species. It would be interesting to search for a further extension of Polya theory within the context of rational combinatorics introduced in [5,6] based on the previous work [12] and further discussed in [7]. One should obtain a generalization of Polya theory in which finite sets are replaced by finite groupoids [3].

Acknowledgment

Thanks to Edmundo Castillo, Federico Hernandez, Eddy Pariguan, Sylvie Paycha, and Domingo Quiroz.

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[16] Rota, G.-C., 1995, Gian-Carlo Rota on Combinatorics, J. Kung, ed., Birkhauser, Boston and Basel.

[17] Vafa, C., and Witten, E., 1995, On orbifolds with discrete torsion, J. Geom. Phys., 15, pp. 189-214.

[18] Zeilberger, D., 2008, "Enumerative and Algebraic Combinatorics", The Princeton Companion to Mathematics, T. Gowers., ed., Princeton University Press, Princeton.

Rafael Diaz

Instituto de Matematicas y sus Aplicaciones, Universidad Sergio Arboleda, Bogota, Colombia

ragadiaz@gmail.com

Hector Blandin

Departamento de Matema ticas Puras y Aplicadas, Universidad Simon Bolivar, Caracas, Venezuela

hectorblandin@gmail.com

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Author: | Diaz, Rafael; Blandin, Hector |
---|---|

Publication: | Global Journal of Pure and Applied Mathematics |

Article Type: | Report |

Geographic Code: | 3VENE |

Date: | Dec 1, 2009 |

Words: | 4468 |

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