Unspecified Journal Volume 00, Number 0, Pages 000-000 S ????-????(XX)0000-0 COUNTING SPECIAL MONOMIALS CHRIS MONICO AND MARA D. NEUSEL Abstract.Let ae : G ,! GL(n, F) be a permutation representation of a fin* *ite group G. In this paper we study the number of orbits of special monomials of G acting on the polynomials in n variables via ae. We give formulae * *for several crucial families of groups, for direct sums of representations, * *as well as for vector invariants. In addition we give two algorithms for arbitr* *ary permutation groups, one relying on the geometry of G acting on V = Fn, t* *he other relying on the representation theory of the symmetric groups. 1.Introduction Let ae : G -! GL(n, F) be a faithful permutation representation of a finite g* *roup G over a field F. The representation ae induces a natural action of G on the ve* *ctor space V = Fn of dimension n, thus on the dual space V *, and hence on the ring * *of polynomial functions F[V ]. Our interest is focused on the subring of invariants F[V ]G = {f 2 F[V ]|gf = f 8g 2 G}, which is a finitely generated F-algebra by a classic result due to E. Noether, * *see Theorem 2.1.4 in [4]. Let x1, . .,.xn be the basis of V *being permuted by the G-action. Let xI = xi11. .x.inn2 F[V ] be a monomial. We order the exponent sequence I in non- increasing order and summarize this in its associated ordered partition ~(I) = (~1(I) . . .~n(I)). Then xI is called special if (1) ~n(I) = 0 and (2) ~i(I) - ~i+1(I) 1, for i = 1, . .,.n - 1. It is known that the ring of invariants F[V ]G of a permutation representation * *is generated by the orbit sums of special monomials and the top elementary symmetr* *ic function sn = x1. .x.n, see Theorem 3.4.2 in [4]. The maximal degree of an alge* *bra generator of F[V ]G in a minimal generating set is called the Noether number. It is denoted by fi(F[V ]G ). By the above mentioned theorem we find that ae` ' oe fi(F[V ]G ) max n2, n ____________ Received by the editors July 27, 2008. 2000 Mathematics Subject Classification. Primary 13A50, Secondary 05Axx. Key words and phrases. Invariant Theory of Finite Permutation Groups, Specia* *l Monomials, Embedding Degree, Permutations, Stirling Numbers, Representation Theory of the * *Symmetric Groups. The second author wants to thank Egon Schulte and Vic Reiner for interesting* * discussions. Oc0000 (copyright holder) 1 2 CHRIS MONICO AND MARA D. NEUSEL for any permutation group G, see Corollary 3.4.3 in [4]. This degree bound is s* *harp as the defining representation of the alternating group An shows, see Example 1* * in Section 3.4 in [4]. Degree bounds are an ongoing theme in the Invariant Theory of Finite Groups with many results for large families of group and representations, see [3] for * *a survey in these matters. In contrast, not much is known about the number of algebra generators of a ring of polynomial invariants, i.e., the embedding degree of the algebra F[V ]G denoted by emb - deg(F[V ]G ). In this paper we want to study the embedding degree of F[V ]G for permutation representations by counting the number NG,V of G-orbits of special monomials. This gives then an upper bound on the embedding degree. We note that the trivial monomial 1 = x01. .x.0nis a special monomial. Proposition 1.1. Let ae : G ,! GL(n, F) be a faithful permutation representatio* *n of a finite group G. Let H G be an arbitrary subgroup. Then the embedding degree is bounded above by the number of orbits of special monomials under the action * *of H: emb - deg(F[V ]G ) NG,V NH,V N{e},V. Proof.Obvious. Even though it is not expected that NG,V ever gives a sharp upper bound on the embedding degree, the number of special G-orbits has the advantage of being independent of the ground field. (This follows because the Poincar'e series of * *F[V ]G is independent of the choice of the ground field F whenever G acts by permutati* *ons, see Proposition 3.2.2 in [4].) In contrast, the embedding degree depends on the choice of F as the 3-fold regular representation of Z=2 shows: The embedding degree of the associated ring of invariants is 9, unless the characteristic of * *F is even. In that case the embedding degree is 10. Furthermore, the study of NG,V is interesting in its own right as this number has intriguing combinatorial proper* *ties as we will see in the following. 2.Special Monomials in F[V ] We start by looking at the trivial group, i.e., we count the number N{e},Vof special monomials in the ambient polynomial ring F[V ] itself thus obtaining a * *uni- versal upper bound for NG,V for an n-dimensional permutation representation of any group G. For this denote by P (n, k) the number of distinct maps from a set with n ele* *ments onto a set with k elements. We note that Xk ` k' P (n, k) = kP (n - 1, k) + kP (n - 1, k - 1) = k!S(n, k) = (-1)k-j jn, j=1 j where S(n, k) is the Stirling number of second kind. Proposition 2.1. The number of special monomials in F[V ] is given by Xn Xn N{e},V= k!S(n, k) = P (n, k). k=1 k=1 Proof.Let S = {x1, . .,.xn} and let T = {0, . .,.k - 1}. Then P (n, k) counts t* *he number of special monomials xI with k different exponents. SPECIAL MONOMIALS 3 Pn We will denote S(n) = k!S(n, k). Note that S(n) counts the total number k=1 of ordered partitions of a set with n elements which coincides with the number * *of surjective maps with a domain with n elements. One may compute S(n) recursively by S(1) = 1 n-1X`n' S(n) = 1 + S(n - i), i=1 i see Page 146 [6]. The first few values of S(n) beginning at n = 1 are 1, 3, 13, 75, 541, 4683, 545835, 7087261, . ... This is sequence A000670 in [5]. Remark 2.2. In Lemma 2 of [1] we find a different formula for the number of spe* *cial monomials in F[V ], namely n 1 fifi N{e},V= _d__dxn___2f-iexfi. x=0 Recall (from, e.g., Page 34 of [6]) that a generating function for the Stirling* * numbers of second kind is given by X xn 1 S(n, k)___= __(ex - 1)k n k n! k! for k 2 N0. Thus n fi fi k!S(n, k) = _d__dxn(ex - 1)kfifi. x=0 Hence we recover the generating function for S(n), see Page 146 [6], Xn Xn dn fifi dn 1 fifi S(n) = k!S(n, k) = ____n(ex - 1)kfifi= ____n_____xfifi. k=1 k=1dx x=fi0 dx 2 - e x=0 Remark 2.3. We note that by Proposition 1.1 we obtain a universal bound on the embedding degree valid for any permutation group: Xn emb - deg(F[V ]G ) NG,V N{e},V= P (n, k) = S(n). k=1 However, this bound is never sharp: Assume it were for some group G n. Then every special monomial would be an invariant. Thus the basis elements x1, . .,.* *xn are invariant. Therefore G = {e} is the trivial group. Its ring of invariants i* *s F[V ] which is generated by the n forms x1, . .,.xn and thus has embedding dimension * *n. 3.Special Orbits in F[V ]G In this section we will derive a formula for the number of special G-orbits f* *or an arbitrary permutation group G that makes use of the geometry of the G-action on the dual space V *. We illustrate the result with the calculation of the number* * of special n-orbits. Let ae : G ,! GL(n, F) be a faithful permutation representation of a finite g* *roup G. Then ae(G) n where we identified the symmetric group with its image in the general linear group under the defining representation. 4 CHRIS MONICO AND MARA D. NEUSEL Proposition 3.1. Denote by ~G,V(k) the number of elements g 2 G such that the dimension of the fixed field V gis exactly k: ~G,V(k) = |{g 2 G| dimFV g= k}|. Then n X NG,V = _1_|G|~G,V(k)S(k). k=1 Proof.For any two integers a, b, a b, we set [a, b] = {a, a + 1, a + 2, . .,.* *b} Z. For 1 k n we let Pn,k= {_ : [1, n] -! [0, k - 1]|_ is surjective} be the set of surjective maps from [1, n] to [0, k - 1]. Thus |Pn,k| = P (n, k)* *. Set Pn = [nk=1Pn,k. Our group G acts on V *by permuting the variables x1, . .,.xn ig-1(1) ig-1(n) g(xi11. .x.inn) = xi1g(1).x.i.ng(n)= x1 . .x.n . Thus we might as well consider G as a group permuting the set [1, n]. We define* * a G-action on the set Pn by G x Pn -! Pn, (g, _) 7! _ O g-1. Furthermore, the set Pn is in one-to-one correspondence with the set of special monomials in F[x1, . .,.xn] by Pn -! {m = xi11. .x.inn2 F[V ]|m special}, _ 7! x_(1)1x_(2)2.x._.(n)n. By construction this correspondence is G-equivariant, i.e., the diagram _ - ! x_(1)1x_(2)2.x._.(n)n # # -1(1))_(g-1(2)) _(g-1(n)) _ O g-1 - ! x_(g1 x2 . .x.n commutes. Thus the number of special G-orbits NG,V is equal to the number of orbits in Pn under the G-action. For g 2 G let Fix(g) denote the number of fixed points of g acting on Pn Fix(g) = |{_ 2 Pn|_ O g-1 = _}|. Then by Burnside's Lemma X NG,V = _1_|G| Fix(g). g2G For a fixed g 2 G, notice that _ O g-1 = _ if and only if (?) _(g-1m) = _(m) for all m 2 [1, n]. The number of independent equations in the System (?) is precisely the number of cycles in the disjoint cycle representation of g (consi* *dered as a permutation of [1, n] and counting 1-cycles). So if c(g) is the number of * *such cycles, it follows that c(g)X c(g)X Fix(g) = P (c(g), j) = j!S(c(g), j) = S(c(g)). j=1 j=1 SPECIAL MONOMIALS 5 Set ~G,V(k) = |{g 2 G|c(g) = k}| to be the number of elements g 2 G which have exactly k disjoint cycles. Then X 1 Xn NG,V = _1_|G| Fix(g) = ___ ~G,V(k)S(k). g2G |G|k=1 Note that ~G,V(k) counts the number of elements g 2 G such that the dimension of the fixed field V gis exactly k. Example 3.2. We consider the defining representation of the symmetric group n. An element oe 2 n has cycle type t(oe) = (1m1, 2m2, . .,.nmn ) if there are mi cycles of length i in its decomposition into disjoint cycles. * *Thus such an element has a fixed field V oeof dimension m1+ . .+.mn. Furthermore, the conjugacy class of oe has |C n(oe)| = __________n!___________1m1m m m 1!2 2m2! . .n.n mn! elements. Note that 1m1 + 2m2 + . .+.nmn = n by definition. Combining these formulas gives Xn _ X ! N n,V = 1_n! |C n(oet)|S(k) k=10 oet 1 Xn B X n! C = 1_n! B@ _______________________m1m2mnCAS(k). k=1 1m1+2m2+...+nmn=n1 m1!2 m2! . .n. mn! m1+...+mn=k Indeed, the number of special n-orbit was already calculated by G"obel in Lemma 1 of [1]: The number of special n-invariant orbits is just 2n-1. This coincides with the number of descending special monomials xi11. .x.inn, i1 i2 . . .in, since every n-orbit contains exactly one descending special monomial. Thus our formula above is just equal to 2n-1. Remark 3.3. We note that NG,V N n,V, i.e., N n,V is a universal lower bound for the number of special G-orbits. 4. Direct Products and Finite Abelian Groups In this section we will illustrate Proposition 3.1 with a few examples of rep* *re- sentations of (mainly abelian) groups. Moreover, we will develop a formula for * *the number of special orbits in direct sums of representations. We start with an example. Example 4.1 (Regular Representation of Z=4). Consider the cyclic group G = Z=4 of order 4 generated by the cycle (1234) 2 4. Thus our group consists of the elements (1), (1234), (13)(24), and (1432). Remembering that we must be careful 6 CHRIS MONICO AND MARA D. NEUSEL to count 1-cycles, we see that Z=4 has one element with four disjoint cycles, o* *ne element with two disjoint cycles, and two elements with only one cycle. Thus NZ=4,V= 1_4(~G,V(1)S(1) + ~G,V(2)S(2) + ~G,V(4)S(4)) = 1_4(2S(1) + 1S(1) + 1S(4)) = 1_4(2 + 3 + 75) = 20. More generally, we consider the regular representation for an arbitrary finite cyclic group. Example 4.2 (Regular Representation of Z=n). Consider the cyclic group of order n, denoted by Z=n, generated by the cycle (123 . .n.) 2 n. Since a permutation (123 . .n.)j has precisely gcd(n, j) disjoint cycles, we find that ( X OE(n=k), if k|n ~Z=n,V(k) = 1 = 1 j n 0, otherwise. gcd(n,j)=k P Thus NZ=n,V= 1_n OE(n=d)S(d). The first few values, beginning at n = 1 are 1, 2, 5, 20, 109, 784, 6757, 68240, 787477, 10224812, . ... This is sequence A019536 in [5], which counts the number of necklaces of n beads with up to n unlabeled colors. We want to develop a formula for sums of representations. We start with two easy examples. Example 4.3 (Elementary Abelian 2-Groups). Consider the Klein-Four Group K4 4 consisting of the elements (1), (12), (34), (12)(34). Then NK4,V = 1_4(S(2) + 2S(3) + S(4)) = 26. More generally, for an elementary abelian 2-group (Z=2)m 2m generated by the transpositions (2i - 1, 2i), i = 1, . .,.m, we find ( m ~(Z=2)m ,V(k) = k-m , if m k 2m, 0, otherwise. Hence 2m X ` m ' N(Z=2)m ,V= _1_2m S(k). k=m k - m The first few terms of this sequence from n = 1 are 2, 26, 818, 47834, 4488722, 617364026, 117029670578, . ... This is sequence A059516 in [5]. It counts the number of ways n intervals of le* *ngth at least zero can overlap. Example 4.4 (Elementary Abelian p-Groups). Let p be a prime number. Consider an elementary abelian p-group (Z=p)m pm generated by the cycles (p(i - 1) + 1, . .,.pi) for i = 1, . .,.m. If m = 1 we find by the calculations in Example * *4.2: NZ=p,V= 1_p(OE(p)S(1) + OE(1)S(p)). SPECIAL MONOMIALS 7 For m = 2 we have NZ=pxZ=p,V= 1_p2(1 . S(2p) + (p - 1)2S(p + 1) + (p - 1)2S(2)). In general we find Xm ` m' N(Z=p)m ,V= _1_pm (p - 1)k S(mp - k(p - 1)). k=0 k The first few terms of this sequence for p = 3 from m = 1 are 5, 555, 273245, 357071595, 969618315005, 4732462989457035, 376932158851544847* *65, . ... For p = 5 from m = 1 we have 109, 4091403, 1842421318717, 4284436559065961835, 34143676145867329865644189, . ... In general, we obtain the following result for direct sums of representations. Proposition 4.5. Let ae : G ,! GL(n, F) and ae0: H ,! GL(m, F) be permutation representations of finite groups G and H. Denote by V = Fn and W = Fm . Then n+mX_k-1X ! NGxH,V xW = __1___|G||H| ~G,V(l)~H,W (k - l)S(k). k=1 l=1 Proof.We note that k-1X ~GxH,V W (k) = ~G,V(l)~H,W (k - l) l=1 and thus we are done by Proposition 3.1. We illustrate this with the computation for arbitrary finite abelian groups. Example 4.6 (Finite Abelian Groups). Let G = Z=n1x . .x.Z=nm n1+...+nm be generated by the m disjoint cycles (1, . .,.n1), (n1+ 1, . .,.n1+ n2), . .,.(n1+ . .+.nm-1 + 1, . .,.n1+ n2+ . .+.* *nm ). Let g = (g1, . .,.gm ) 2 Z=n1 x . .x.Z=nm . Let Vi be the corresponding subspace of dimension ni. Then dimF V g= dimFV1g1+ . .+.dimFVmgm. We obtain by the preceding proposition X ~G,V(k) = ~Z=n1,V1(l1) . .~.Z=nm ,Vm(lm ). l1+...+lm =k P NZ=n,V= 1_n OE(n=d)S(d). Thus X X NG,V = ___1____n OE(n1=l1) . .O.E(nm =lm )S(k) 1.k.n.ml1+...+lm =k X X = ___1____n . . . OE(n1=l1) . .O.E(nm =lm )S(l1 + . .+.lm ). l1.1.n.m|n1lm |nm The following tables gives some values of NG,V for G = Z=n1 x Z=n2. 8 CHRIS MONICO AND MARA D. NEUSEL ___________|_n_=_1_|__n_=_2_|___n_=_3_|___n_=_4_|______n_=_5_| Z=1 x Z=n | 3 | 8 | 27 | 140 | 939 | Z=2 x Z=n | 8 | 26 | 108 | 668 | 5204 | Z=3 x Z=n | 27 | 108 | 555 | 4092 | 37035 | Z=4 x Z=n | 140 | 668 | 4092 | 34844 | 357308 | Z=5 x Z=n | 939 | 5204 | 37035 | 357308 | 4091403 | Z=6 x Z=n | 7900 | 49496 | 399332 | 4289444 | 54115732 | Z=7 x Z=n | 77979 | 545228 | 4920939 |58243388 | 802679403 | Z=8 x Z=n |885980 |6833768 |68202372 |881517284 |13172347508 | We close this section with a nonabelian example. Example 4.7 (The Dihedral Group D2n of Order 2n). For n 3 let D2n denote the dihedral group of order 2n. It has a natural permutation representation of dimension n as D2n can be considered as the permutation group of the vertices of a regular n-gon. As such it is generated by a rotation (1, 2, . .,.n) and a ref* *lection (1, n)(2, n - 1) . .(.n_2, n+2_2) for n even, resp. (1, n)(2, n - 1) . .(.n-1_2* *, n+1_2) for odd n. See Example 2 in Section 5.8 of [4] for the special case of n = 5. We can partition D2n as D2n = Z=n t Rn where Rn consists of elements of order 2. Having already deduced the necessary information about the disjoint cycle representati* *ons of elements of Z=n in Example 4.2, it remain only to do the same for elements in Rn. But since these are elements of order 2, the required information can be deduced by counting the number of fixed points of each elements of Rn, which wi* *ll depend on the parity of n. The end result is ( 1 _NZ=n,V+ 1_S(n+1_), for n odd, ND2n,V= 21_ 21_n_2 1_ n+2_ 2NZ=n,V+ 4S(2 ) + 4S( 2 ),for n even with NZ=n,Vfrom the previous example. The first few terms, from n = 1 are 1, 2, 4, 14, 61, 414, 3416, 34274, 394009, 5113712, . ... This is Sequence A019537 in [5]. 5.Regular Representations and Vector Invariants If G acts diagonally on m copies of V then ` ' ~G,V m(k) = ~G,V _k_m We want to find the function ~G,V for the regular representation of an arbitrary finite group G, FG. If g 2 G, then G is a cyclic subgroup. Thus FG restri* *cted to gives the |G : |-fold regular representation of . Hence fifiae fifoei ~G,V(k) = fifig 2 G | |g| = |G|_kfifi. Thus X 1 X fifiae |G|oefifi NG,V = _1_|G| ~G,V(k)S(k) = ___ fifig 2 G||g| = ___fifiS(k) k||G| |G|k||G| k SPECIAL MONOMIALS 9 for the regular representation. Thus for the m-fold regular representation we g* *et |G|mX NG,V = _1_|G| ~G,V m(k)S(k) k=1 |G|mX ` k' = _1_|G| ~G,V __ S(k) k=1 m |G|X = _1_|G|~G,V(l)S(ml). l=1 More generally we find for arbitrary vector invariants: Proposition 5.1. Let ae : G ,! GL(n, F) be a faithful representation of a finite group G. Let V = Fn, and set ae1 = ae. Then the m-fold vector invariants of ae * *are given by the diagonal action of G on m copies of V . We find X|G| NG,V m= _1_|G|~G,V(k)S(mk). k=1 Proof.This is a straightforward generalization of the case of vector invariants* * of the regular representation which we discussed above. 6. Special Orbits in F[V ]G : an Algorithm In this section we show how to use the results from the preceding sections to create another algorithm that counts the number of orbits of special monomials * *for an arbitrary permutation group which relies on the representation theory of the symmetric group. Let ae : n ,! GL(n, F) be the defining representation of n. Let V = Fn. Then we define the function ~ : n -! N, g 7! dimV g. Then for any subgroup G n we obtain ~G,V(k) = |~-1(k) \ G|. Lemma 6.1. Note that this implies |G|NG,V N n,V| n|, since ~ is nonnegative. Thus we obtain a relative bound for the number of speci* *al orbits n-1n! NG,V N n,V| n : G| = 2_____|G|. Proof.We simply compute Xn Xn |G|NG,V = ~G,V(k)S(k) ~ n,V(k)S(k) = | n|N n,V. k=1 k=1 10 CHRIS MONICO AND MARA D. NEUSEL We observe that ~ is a class function. Even more, ~ is constant on the sets of elements in n consisting of the same number of disjoint cycles. Thus we can wr* *ite X 1 X NG,V = _1_|G||Cg(G)|S(c(g)) = ___ |Cg( n) \ G|S(c(g)) g |G| g where the sum runs over a set of representatives of the conjugacy classes, Cg(G* *) is the conjugacy class of g 2 G, and c(g) is the number of disjoint cycles in g. T* *he disadvantage of this formula is that it requires precise knowledge about the em* *bed- ding G ,! n, because one needs to find the values of |Cg( n) \ G|. Therefore, * *we propose the following: Proposition 6.2. X 1 X NG,V = _1_|G| S(dim V g) = ___ S(~(g)). g2G |G|g2G Proof.Follows immediately from Proposition 3.1. Since our calculations are characteristic free, we might as well assume we are working over the complex numbers, thus we can write the class function ~ as a linear combination of the characters of the full symmetric group. Let pi(n) be * *the number of partitions of n into i nonnegative integers. Then the total number of conjugacy classes in n is equal Xn C = pl(n) l=1 (i.e., the total number of partitions into nonnegative integers). Thus the char* *acter table of n is a C x C complex invertible matrix, call it M. Thus if we write t* *he function ~ as a linear combination of the irreducible characters Oi of n ~ = a1O1 + . .+.aC OC we find 2 3 2 3 a1 n 66a27 6n - 17 64..77= M-1 66 . 77 . 5 4 ..5 aC 1 where in the column vector on the right each i appears pi(n) times. Thus once we have the character table of n, we can compute the function ~ and hence NG,V for any subgroup G n. We will study this in more detail in a succeeding paper, [* *2]. For now, we want to illustrate this process with the calculations for n = 2, . * *.,.6. Example 6.3 ( 2). The following table contains in character table of 2 and, in the last row, the values of the function ~ on the conjugacy classes: ____|(1)_|(12)_ We obtain the two linear equations _O1_|_1__|_1__ _O2_|_1__|_-1_ a1 + a2 = 2 and a1 - a2 = 1. ~ |2 | 1 Thus a1 = 3=2 and a2 = 1=2. Example 6.4 ( 3). The following table contains in character table of 3 and, in the last row, the values of the function ~ on the conjugacy classes: SPECIAL MONOMIALS 11 ____|(1)_|(12)_|(123)_ Direct computation yields _O1_|_1__|_1__|__1___ a1 = 5=3, a2 = 2=3, and a3 = 1=3. _O2_|_1__|_1__|__-1__ _O3_|_2__|_-1__|_0___ ~ |3 | 2 | 1 Example 6.5 ( 4). The following table contains in character table of 4 and, in the last row, the values of the function ~ on the conjugacy classes: ____|(1)_|(12)_|(123)_|(12)(34)(|1234)_ _O1_|_1__|_1__|__1___|___1____|__1___ _O2_|_1__|_-1__|_1___|___1____|_-1___ _O3_|_2__|_0__|__-1___|__2____|__0___ _O4_|_3__|_1__|__0___|__-1____|_-1___ _O5_|_3__|_-1__|_0___|__-1____|__1___ ~ |4 | 3 | 2 | 2 | 1 Direct computation yields a1 = 25=12, a2 = 1=12, a3 = 1=6, a4 = 3=4, and a5 = -1=4. Example 6.6 ( 5). The following table contains in character table of 5 and, in the last row, the values of the function ~ on the conjugacy classes: ____|(1)_|(12)_|(123)_|(12)(34)(|1234)(|123)(45)(|12345)_ _O1_|_1__|_1__|__1___|___1____|__1___|___1_____|__1____ _O2_|_1__|_-1__|_1___|___1____|_-1___|___-1____|__1____ _O3_|_4__|_2__|__1___|___0____|__0___|___-1____|__-1___ _O4_|_4__|_-2__|_1___|___0____|__0___|___1_____|__-1___ _O5_|_6__|_0__|__0___|__-2____|__0___|___0_____|__1____ _O6_|_5__|_1__|__-1___|__1____|_-1___|___1_____|__0____ _O7_|_5__|_-1__|_-1___|__1____|__1___|___-1____|__0____ ~ |5 | 4 | 3 | 3 | 2 | 2 | 1 Direct computation yields a1 = 137=60, a2 = -1=20, a3 = 4=5, a4 = 2=15, a5 = -3=10, a6 = 1=4, and a7 = -1=12. Example 6.7 ( 6). The following table contains in character table of 6 and, in the last row, the values of the function ~ on the conjugacy classes: 12 CHRIS MONICO AND MARA D. NEUSEL _____|(1)_|(12)_|(123)(|12)(34)(|1234)(|123)(45)(|12345)(|12)(34)(56)(|123)(456* *)(|1234)(56)(|123456)_ __O1_|_1__|1__|__1___|__1____|__1___|___1____|__1____|____1______|___1_____|___* *1_____|__1_____ __O2_|_1__|-1__|_1___|__1____|_-1___|___-1____|_1____|____-1_____|___1_____|___* *1_____|__-1____ __O3_|_5__|3__|__2___|__1____|__1___|___0____|__0____|____-1_____|___-1_____|__* *-1_____|_-1____ __O4_|_5__|-3__|_2___|__1____|_-1___|___0____|__0____|____1______|___-1_____|__* *-1_____|_1_____ __O5_|10_|_2__|__1___|__-2____|_0___|___-1____|_0____|____-2_____|___1_____|___* *0_____|__1_____ __O6_|10_|_-2__|_1___|__-2____|_0___|___1____|__0____|____2______|___1_____|___* *0_____|__-1____ __O7_|_9__|3__|__0___|__1____|_-1___|___0____|__-1___|____3______|___0_____|___* *1_____|__0_____ __O8_|_9__|-3__|_0___|__1____|__1___|___0____|__-1___|____-3_____|___0_____|___* *1_____|__0_____ __O9_|_5__|1__|__-1___|_1____|_-1___|___1____|__0____|____3______|___2_____|___* *-1_____|_0_____ _O10_|_5__|-1__|_-1___|_1____|__1___|___-1____|_0____|____-3_____|___2_____|___* *-1_____|_0_____ _O11_|16_|_0__|__-2___|_0____|__0___|___0____|__1____|____0______|___-2_____|__* *0_____|__0_____ ~ | 6 |5 | 4 | 4 | 3 | 3 | 2 | 3 | 2 | * *2 | 1 Direct computation yields a1 = 289=120, a2 = 3=40, a3 = 7=8, a4 = -1=8, a5 = -1=4, a6 = 1=12, a7 = 7=40, a8 = 7=40, a9 = 5=24, a10= -1=8, and a11= -2=15. References 1.M. G"obel, On the number of special permutation-invariant orbits an* *d terms, Applicable Al- gebra in Engineering, Communication and Computing 8 (1997), 505-509. 2.C. Monico and M. D. Neusel, Counting Special Monomials II, in prepa* *ration. 3.M. D. Neusel, Degree Bounds. An Invitation to Postmodern Invariant * *Theory, Topology and its Applications 154 (2007), 792-814. 4.M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Mathe* *matical Surveys and Monographs, Volume 94, AMS, Providence RI 2002. 5.N. J. A. Sloane, The On-Line Encyclopedia of Integ* *er Sequences, http://www.research.att.com/ njas/sequences/, 2008. 6.R. Stanley, Enumerative Combinatorics Volume I, Cambridge Studies i* *n Advanced Mathe- matics 49, Cambridge University Press, Cambridge 1997. Department of Mathematics and Statistics, MS 1042, Texas Tech Univer* *sity, Lub- bock, Texas 79409 E-mail address: C.Monico@ttu.edu and Mara.D.Neusel@ttu.edu