I nternat. J. Math. & Math. Si.
Vol. 5 No. 4 (1982) 745-762
745
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
CHONG-YUN CHAO
Department of Mathematics
University of Pittsburgh
Pittsburgh, PA 15260
CAROLINE I. DEISHER
Department of Mathematics
Indiana University of Pennsylvania
Indiana, PA
iReceived
ABSTRACT.
By using
15705
June 31, 1981 and in revised form June 7, 1982)
Plya’s
theorem of enumeration and de Bruijn’s generalization of
Plya’s
theorem, we obtain the numbers of various weak equivalence classes of funcD
D
tions in R relative to permu.tation groups G and H where R is the set of all functions from a finite set D to a finite set
R, G acts on D and H acts on R.
We present
an algorithm for obtaining the equivalence classes of functions counted in de
theorem, i.e., to determine which functions belong to the
same
BruiJn’s
equivalence class.
also use our algorithm to construct the family of non-isomorphic
fm-graphs
We
relative
to a given group.
KEY WORDS AND PHRASES.
algorithm,
Enumerations, equivalence classes of functions on finite sets,
fm-graphs.
980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
I
Primary.
05A15
Secondary.
05C30
INTRODUCTION
Motivated by Carlitz’s work in [i] on the invariantive properties over a finite
field
K,
Cavior
([2],[3]) and Mullen ([4],[5],[6],[7]) studied several families
of
C.Y. CHAO AND C.I. DEISHER
746
These equivalence relations can be
equivalence relations of functions from K into K.
Let
described in more general forms as follows:
R
D
on
D
be the set of all functions from
D
(I)
Let
H,
and
H
and
be a permutation group acting on
f,g c R
D.
(a)
When
g(d)
H
for every
G,
When
G
(c)
When
i.e.,
H,
G
H
f,g
g
i.e.,
u-lfu
and
g
equivalent to
and a
H
T
G
relative to
T-Ifo
such that
g,
f
is said to be right equivalent to
f
is said to be left equivalent to
g
g.
fu
is the identity group,
relative to
R.
There are three subfamilies:
D.
d
is the identity group,
relative to
(b)
o c G
be a permutation group acting
G
is said to be weakly
f
if and only if there exist a
T-if(d)
i.e.,
R,
into
n},
{l,2,...,m}, R- {1,2
D--
T-If
g
g.
g
is said to be similar to
f
g,
relative to
G.
(II)
and
H,
Let
R
D
f
is said to be strongly equivalent to
if and only if there exist a
G
o
and
T E
H
relative to
g
such that
fu
g
G
and
f--g.
Clearly, all of these relations above are equivalence relations.
One of
Cavior’s and Mullen’s main results was to obtain the number of equivalence classes of
functions over
Plya’s
K
relative to symmetric groups, and to cyclic groups.
theorem of enumeration and de
BruiJn’s generalization
of
Here by using
Plya’s theorem,
we
shall point out that the numbers of various weak equivalence classes of functions in
R
D
relative to
G
and
H
can be obtained.
We shall present an algorithm for ob-
taining the equivalence classes of functions counted in de Bruijn’s theorem, i.e., to
determine which functions belong to the same equivalence class.
associate each function with its incidence matrix.
Our method is to
Various weak equivalence re-
lations correspond to products of matrices, and from the entries of the incidence
matrices, equivalent functions can be obtained.
cycle indices of the permutation groups.
family of non-lsomorphic
fro-graphs
Our algorithm does not use the
We use our algorithm to construct the
relative to a given group.
equivalence classes do not appear to be obtainable from
theorems.
Cavior, in
Plya’s
The numbers of strong
and de
BruiJn’s
[2], obtained the number of strong equivalence classes relative
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
We apply our algorithm to strongly equivalent functions.
to the symmetric groups.
With the help of
Plya’s
747
and de
BrulJn’s theorems,
our algorithm enables us to deter-
strong equivalence classes relative to some subgroups of the sym-
mine the numbers of
metric groups.
2.
THEOREMS OF POLYA AND DE BRUIJN.
Let
{1,2
D
be a permutation group acting on a set
G
,m}.
Since every
permutation can be uniquely written as a product of disjoint cycles, the cycle index
of
G
field of rational numbers and
xlxj
for
xjx i
b
I
I<I
G
+/-s the order of
and
b
u
for
disjoint cycle decomposition of
THEOREM i.
finite set
D
into a finite set
a function from
R
g
Q,
[
PG( rR wCr),
rcR
he defined on
with
g(d)
f(d)
D
R
Consequently,
F
where each
is
in the
w
be
is a commutative ring with an identity con-
G
W(F),
i
D,
be a permutation group acting on
G
(This relation is an equivalence relation.
total patterns, denoted by
x m
m
be the set of all functions from a
and a relation
{F},
b
I x 2
2
D
R
Let
if and only if there exists a
disjoint equivalence classes
m:
1,2,...,m.
i
R’ where R’
into
raining the rational numbers
f
R,
is the
Q
is the number of cycles of length
i
(Plya [8],[9],[i0]).
b
where
x
m
1,2
i,j
PG (’2’ "’’’) "7 G x I
where
Q[Xl,X2
is defined as the following polynomial in
D
R
such that
for every
d
D.
is partitioned into
is called a
pattern.)
Then the
F
WCF)
F
where
PG
(w(r))2,..., [ (wCr))k,...)
is the cycle index relative to
(1)
reR
G.
If
w(r)
I
for every
r e R,
then
the number of total patterns is
IXF W(F)l P(IP,.I,IP-I,...,IRI,...)
where
IRI
is the cardlnallty of
THEOREM 2.
(2)
R.
(de BrulJn [11],[12]).
Let
D
R
be the set of all functions from a
C.Y. CHAO AND C.I. DEISHER
748
D
finite set
f
and a relation
R
a permutation group acting on
o e G
if and only if there exist a
g
every
be defined on
e H
and a
Consequently,
{F}, where each F
tltloned into disjoint equivalence classes
be
such that
rg(d)
f(od)
with
(This relation is an equivalence relation.
D.
d
D
R
H
D,
be a permutation group action on
G
R,
into a finite set
R
D
is called a
for
is par-
pattern.)
Then the number of total patterns is
z--
[pG(
evaluated at
H
If
z
z
I
cardlnallty is
Let
D
be defined on
tion
D
"D
D
valence relation.
Consequently,
F
be a permutation group acting on
G
such that for every
o e G
where each
with
is called a
o
-I f(od)
D
D
of
o
for
for every
f
d e D.
D,
g
whose
and a relaif and only
(This is an equi-
Then the number of total patterns is
m
i=l
J cj
j i
The number of equivalence classes in
of)
fo
and the number of
c.l
(4)
in the disjoint cycle decomposition
i
is
i=lH (j i
D
c
j
D
f e D
m
of
fo
3.
g e D
and
D
D
theorem.
1,2,...,m.
i
such that
(3)
is partitioned into disjoint equivalence classes
is the number of cycles of lenght
c.
f
g(d)
pattern.)
I
oeG
where
)]
Plya’s
be the set of all functions from a finite set
if there exists a
{F},
3(z3+z6+...)
then (3) is (2) in
R,
group acting on
into itself,
m
e
0.
2
is the identity
THEOREM 3.
Zl+Z2 +’’’, e 2(z2+z4+’’’)
PH(e
z3
cj)
i
is
D
I
(number of functions
f
such that
For details, see [7].
AN ALGORITHM.
Let
G
and
H
be permutation groups acting on
{l,2,...,n} respectively.
tion in
D
R
said to be
relative to
D
{l,2,...,m} and R
For convenience, we shall call the weak equivalence relaD
are
f and g in R
G and H the G-H-relation, i.e
G-H-related if and only if there exists a
o e G
and a
T e
H
such that
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
T
-i
g(d)
f(od)
d e D.
for every
Clearly, it is an equivalence relation, and
D
R
Let
G*
is partitioned into disjoint classes each of which is called a
be the
n
permutation group corresponding to
n
all
(O,l)-matrices A
n
m
G,
permutation group corresponding to
m
m
(aij)
H,
related if and only if there exist a
P e G*
A
H,
be the set of
G*-H*-
are said to be
Q e H*
and a
be the
consists of exactly one
B
and
H*
I
and
A
where each row of
G-H-class.
G*-- G,
i.e.,
H*__
i.e.,
Two matrices
i and all other entires are zero.
749
-I
PAQ
such that
B.
I
Clearly, this relation is an equivalence relation called a G*-H*-relation, and
G*-H*-
is partitioned into disjoint equivalence classes each of which is called a
class.
[13],
Similar to Lemma 1 in
THEROEM 4.
for
i
ai,f(i)
is a bijective map, and
(2)
preserves the
for
(i) Clearly,
bi,g(i
bi,g(i
i
for
for
i
IRDI III
m
n
Let
and a
T
ai,f(i)
Then
b
i,
P
m,
and
f
e H
1
-if(i)
Let
and
i
(Pij)
respectively.
i
Hence,
belong to the same
g
such that
bi,g(i
for
and
B
If
B,
A
then
ai,f(i)--
Since
is injective.
I
(f)
g,
for
i
A
1,2
(aij)
-
(qij)
n(g)
and
B
(bij)
and all other entries 0.
,m
1,2,...,m and all other entries
i
Q--
T-Ifo
G-H-class, i.e., there exist a
0.
be the permutation matrices corresponding to
By using the properties of permutation matrices, we have
(PAQ-I) ij
for all
(g)
is bijective.
(2)
with
and
1,2,
g.
i.e., f
I.
(aij) with ai,f(1)-(bij) with
A
1,2,...,m and all other entries 0.
i
with
G*-H*-relation in
(f)
and
0,
(aij)
Then
and the
Let
is well defined.
and all. other entries
1,2,...,m
i
D
R
G-H-relation in
A
A where
and all other entries 0.
1,2,...,m,
i
(f)
be defined by
(i)
PROOF.
i
D
R * I
:
Let
we have
It
s
1,2,...,m and j
Pisastqjt
Piuauvqjv
1,2,...,n with
i
auv
u
(5)
aoi,;J
and
Tj
v.
But since all
C.Y. CHAO AND C.I. DEISHER
750
--0
aoi,T j
T-if(ol)
Hence,
except
(PAQ -I)
and
PAQ
-I
B,
G*
n
say
-IA
and
A
and
B
0, except
n
and
f
-
A
Q e H*
and a
ai, f(i)
(PAQ-I)ij
for
for
(B)ij
1,2,...,m,
i
PAQ
(bij)
and
g,
f
(aij)
I
in
with
I
ai, f(i)
i
-li, m-lf(i)
B
in
I.
Let
and
to
for
P--
(plj)
respectively.
From Theorem 4, for each
and g
are
G-H-related.
(Qij)
Q
and
Then
B e I
PAQ
-I) iJ are
T-If(oi).
g(i)’
G-H-class.
and all other entries
0
n(f)
(A
0
z e H,
and every
G
o
1,2,...,m and all other entries
i
exists,
there exists a unique
i- 1,2,...,m
Now for every
f).
is called the incidence matrix of
a
for
f
T-if(ol)
belong to the same
g
RD
exist a
for i- 1,2,...,m.
1,2,...,n,
and
J
or
1
bl,g(i
J
and
f(i)
]
-i
all (PAQ
ai,rJ’
except
m
n
is biJective,
i.e.,
0,
are
n
(PAQ-I)ij
From our Theorem 4, we know that for each
A
G*-H*-class, then there
Since
(5)
1,2
i
T-ifo
i.e.,
B.
1,2,...,m,
i
for
0.
1,2,...,m and all other entries
i
-i
Since by
g.
J
i.e.,
belong to the same G*-H*-class.
B
such that
I
for
i
fCol),
TJ
1,2,...,m,
i--
belong to the same
Also, all the entries of B
Since
for
i,m-lf(oi)
and
Conversely, if
P
i
aGi,f(ol)
determine a matrix
be the permutation matrices corresponding
-I
B
and
A and
there exists a unique
B
are
n-l(B)
g
G*-H*-related.
in
R
D
and
f
Consequently, we have the following algorithm for obtaining
all equivalence classes, i.e., for determining which functions are in the same equivalence class:
Step
i.
Select any
D
f e R
and write
Ste_.
For every
a
e G and every
_ii, T -If (l)
m, f(m)
z e H,
compute
a-Im, -if (m)
a-I 2,-If(2)
Each computation determines a function in
f
a
a2,f(2)
al,f(1)
D,
R
and the equivalence class containing
consists of these distinct functions.
Step 3.
Select a function in
obtained in Step 2.
D
R
which is not a member of the equivalence class
Repeat Steps i and 2.
Continue the process until every function
751
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
in
D
R
appears in an equivalence class.
EXAMPLE i.
{TI
(i),
T2
Let
{c
G
(12)}
(i),
I
{1,2%.
R
act on
(23)}
u2--
D
act on
{1,2,3} and H
Then the cycle indices of
G
and
H
are,
respectively,
i
Pc(Xl,X2,X3)
(x3 +
I
XlX2)
and
1(2
Yl + Y2 )"
PH(YI’Y2
D
Therefore, by using (3), the number of weak equivalence classes in R
8 functions in
D
R
is 3.
If the
are given as
fl f2 f3 f4 f5 f6 f7 f8
i
i
i
2
I
2
2
2
i
i
2
I
2
I
2
2
I
2
i
I
2
2
i
2
then the algorithm can be used to determine the 3 classes so that
RD {fl,f.8} k {f2’ f3’ f6’ f7 } U {f4,f5}.
4.
Let
APPLICATIONS.
We consider the weak equivalence classes in RD
D
f}.
and C(f)
f
R
{(u,T) e G H;
T-Ifu
of the product group
THEOREM 5.
D
R
G
H.
Clearly,
C(f)
The cardinality of the weak equivalence class
Hence,
G and H.
is a subgroup
G x H.
relative to the groups
group
relative to the groups
II
G
H
and
is equal to the index of
containing
C(f)
f
in
in the product
divides
The proof is not difficult and hence is omitted.
COROLLARY 5.1.
in
D
R
The cardinallty of the right equivalence class
relative to the group
G
is
equal to the index of the subgroup
containing
C(f)
f
C.Y. CHAO AND C.I. DEISHER
752
The cardinality of the left equivalence class
COROLLARY 5.2.
D
R
in
m-if
{me H;
f}
H.
in
COROLLARY 5.3.
in
f
containing
D
R
in
o-lfo
{o e G;
C(f)
GI.
The cycle indices for many families of groups are known, e.g., see p. 36 in
A.
[9].
divides
C(f)
INf.
divides
is equal to the index of the subgroup
15]
f
containing
equal to the index of the subgroup
II
Hence,
G
Hence,
G.
is
The cardinallty of the similar class
relative to the group
f}
H
relative to the group
f
In particular, the cycle index of the cyclic group
C
q
of order
on
q
p
points is
q
PCq (Xl’X2’’"’Xq)
#(i)
where
is the
EXAMPLE 2.
q+k
acting on
Euler’s phi-function.
Let
<(123...q)>
G
be the cyclic group generated by
G
be the permutation
R
of the right equivalence classes in
z--
[PG(-I
(
Xq+k)
group acting
be the identity group acting on
N
(123...q)
Then
points.
PG(Xl,X2,... xq,Xq+I,
Let
i q
on
D
{l,2,...,q,q+l,...,q+k
{l,2,...,n}. Then, by
D
R
relative to
(e
--)
m
(6)
Xl).
G
m}
and
H
using (3), the number
is:
n(zl+z2+...+zn+...)
)]Zl=Z2=...=0
q
I
(#(i)n i
nk).
(7)
qiqT
In particular, if
tity group acting on
D
D
relative to
G
1
G
<(1234)>
1
R
D,
is
825.
acts on
D
{1,2,3,4,5} and
H
is the iden-
then by (.7) the number of right equivalence classes in
We show the following:
(a)
There are 25 right equivalence classes each of cardinality 1.
algorithm, the function corresponding to
By using our
753
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
ali
is not changed for
(b)
any
a3i
a2i
GI,
o m
1,2,...,5
i
i
a5j
a4i
and
1,2,...,5.
j
There are 50 right equivalence classes each of cardinality 2.
(1234),
o
2
all
for
i
# J,
3
(13)(24),
a2j
5
1,2,
i,j
(1432)
and
a3i-- a4j
and
4
o
to
i
a5k
5,
1,2,
k
(i)
Applying
we have
a4i
alj
a2i
a3j
a5k
I,
(8)
a3i
a4j
all
a2j
a5k
i,
(9)
a2i
a3j
a4i
alj
a5k
i,
all
a2j
a3i
a4j
a5k
I.
(i0)
and
(Ii)
Since (8) and (9) are the same as (i0) and (ii) respectively, there are
()
5
(c)
50
right equivalence classes each of cardinality 2.
Since by Corollary 5.1 there is no right equivalence class of cardinality
3, the
number of right equivalence classes of cardinality 4 is 825-25-50
Our
results in this example with
in
<(1234)>
I
coincide with the results on p. 113
[12].
EXAMPLE 3.
acting on
q+k
Let
<(123...q)> be the cyclic group generated by
H
points.
Then the cycle index of
R
{l,2,...,q,q+l,
left equivalence classes in
H
{l,2,...,m} and H
D
the identity group acting on
ing on
G
750.
n}.
q+k
is the same as
be the group
(6).
relative to
N
lq [n + iq @(i)km].
H
Let
G
<(12...q)>
Then, by using (3), the number
D
R
(123...q)
N
be
act-
of the
is given by
m
(12)
i>l
In particular, let
be the group
<(1234)>
G
be the identity group acting on
acting on
R
D.
D
{1,2,...,5}
and
H
I
Then the number of left equivalence classes
754
C.Y. CHAO AND C.I. DEISHER
D
D
in
H
relative to
is 782 by using
I
By using our algorithm, the function corresponding to
We show the following:
a15
where
Then with
a
l,z
a
-i
a
2,z
-I
3) -lj 1
a
3) -lj 2
a
3, (z 2) -lj 3
a
4, (.z 2) -lj 4
a
3, (z 3)
a
4, (z 3) -lj 4
a
4
a3, J3
a2’J2
ai’Jl
a
3,
2, (z 2) -lj 2
2, (z
z- I-]3
a
a
i, (z
5
I
’J5
1,2,...,5.
k
J2
I, (_T2) -lj i
a
a
(.1234), we have
T
Jl
I
5
Now consider
a3’J3 a4’J4
a2, J2
are not all 5 for
Jk
as.
a45-
a35
a25
is in an equivalence class by itself
al’Jl
(12), or hy using (3) and (7).
_lj3
a
4, -iJ4
a
5, (.z2) -lj 5
i,
a
5, (z3) -lj 5
i,
a
’J4
i,
5, -i J5
5
’J5
All of the functions corresponding to the above matrices are different and therefore,
Hence, there
the cardinality of the equivalence class is 4.
equivalence
is only one
class of cardinality i and all the others are of cardinality 4, that is, there are
782
i
781
equivalence classes of cardinality 4.
Our results in this example
thus coincide with the results on p. 353 in [4].
Thus, the incidence matrix of an
out-degree at every vertex is i.
to the set of
m x m
que
fm-graph.
Let
X
and
I
that
f -graph if the
m
A labeled directed graph with m vertices is said to be an
B
o
X
2
matrices
G
I.
Conversely, every matrix in
be a permutation group acting on
m
fro-graph
I
determines a uni-
Two
points
are said to be G-isomorphic if and only if there exists a
maps the vertices of
directed edges, i.e.,
a directed edge in
X I.
[oa, ob]
XI
onto the vertices of
is a directed edge in
A G-isomorphism of
X
I
X2,
X
2
and
belongs
o
fm-graphs
o
G
such
preserves the
if and only if
[a,b]
onto itself is said to be an auto-
is
755
EQUIVALENCE CIASSES OF FUNCTIONS ON FINITE SETS
AI
Let
A2
morphlsm of
X I.
spectively.
Then it is well known that
and
P
there exists a permutation matrix
XI
and
G,
c
of
e
for
1,2
i
EXAMPLE 4.
classes of
to
J
i
-I
(13)
i
in the disjoint cycle decomposition
.... ,m.
Let
f3-graphs
<(123)>
G
act on
O
relative to
is,
{1,2,3}.
Then the number of nonisomorphic
by using (13),
I
[(1.3)
3
By using our algorithm, we have the following nonisomorphic
ii.
PAIP
i
is the number of cycles of lenght
i
such that
i.e., the number of nonlsomorphlc
m
eG i=l
c
u e G
is
I
where
2
we may use (4) to count the number of non-
isomorphic classes of fm-graphs relative to
G
re-
points into itself is in one to one cor-
m
fro-graphs,
classes of f -graphs relative to
m
and
are G-isomorphlc if and only if
X
corresponding to a
Since the set of all functions from
respondence with the set of all
XI
be the incidence matrices of
+ (3.1) i + (3-1) i]
f3-graphs
relative
G:
i
I
1
3
2
(ii)
(i)
2
by using
(iii)
(iv)
(viii)
3
(v)
(vl)
(vii)
(ix)
(x)
(xi)
EXAMPLE 5.
{1,2,3}.
3
2
Replace
G
<(123)>
in Example 4
Then the number of nonisomorphic classes of
(13),
1
[(1"3)
3
by the symmetric group
S
relative to
S
f3-graphs
+ 3((1.1) (1"1+2"1) l) + 2(3-1) i]
1
7,
3
3
on
is,
and the nonlsomorphic
C.Y. CHAO AND C.I. DEISHER
756
(relative to
f3-graphs
S
3)
(i), (ii), (iv), (v), (vi), (viii) and (x), i.e.,
are:
(ii) is isomorphic to (iii) by (23), (.vi) is isomorphic to (vii) by (12), (viii) is
isomorphic to (ix) by (13), and (x) is isomorphic to (xi) by (23).
By applying (4), de BruiJn in [12] obtained the number Nm
valence classes relative to the symmetric group
k
m
J
m
S
(e) i--1
k
k
which satisfy
i
are given by
N
I,
I
N
I + 2k2 +
N
3,
2
(k
+mk
n
7,
3
ikl) -I
i
where the first summation is over all m-tuples
tegers
as
m
kj) (ki!
j +/-
N
I k2
(14)
of non-negative in-
k
m
N
The first few values of
m.
47,
N5
19,
4
of similar equi-
N
130.
6
m
Formula (14)
[2, p. 128] concerning the number
gives the answer to the problem posed by Cavior in
of similarity classes relative to the symmetric group.
C.
in
D
D
group on
[2],
On p. 129 in
relative
D.
to’
S
m
Cavior obtained the number of strong equivalence classes
S
and
where
m
{l,2,...,m}
D
S
and
is the symmetric
m
Here, with the help from the theorems of P61ya and de Bruijn, we apply
our algorithm to obtain the following theorems.
THEOREM 6.
Let
,m}
{l,2,
D
where
D
be any permutation group actin 8 on
and
D
R
m
H
<(12)>
be the group
and
H
one function, i.e., the number of strong equivalence classes is
2
Then every strong equivalence class in
Let
PROOF.
f
f
be any function in
R
G
relative to
and
R
is an odd integer,
(aij)
Af
{1,2},
acting on
G
R.
consists of only
m.
be the incidence matrix
with
a2,12
al,i I
and all other entries
0
where
i
m,im
is either i or 2 for
k
1
a
a3,i 3
using our algorithm, the right equivalence class relative to
sists of the functions corresponding to the set of matrices
a _i
0
a
I ,i I
_12 ,i 2
a-I
0
3,i
left equivalence class relative to
G
containing
[Af;
s e G,
Then, by
f
con-
and
and all other entries 0}.
i
a-i
3
1,2,...,m.
k
The
m
H
containing
f
consists of the set of matrices
757
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
{ATf;
H
T e
and
a
i,
entries 0}.
(12).
Since
m
a
i
_i13
{-lil ,T -I i2
3, T
odd, the set
is
a
a
_ii2
2,
I
-i
m,
T
-I
m}
# {i I i2,...,im
i
and the right equivalence class relative to
H
IRI IDI
2
from a finite set
H
D,
is
and
H
con-
{f}.
is
m.
In [8], Theorem 5.3 states:
the domain
G
for
G
to
f
containing
Consequently, the number of strong equivalence classes relative to
IRDI
and all other
i
i
m
Hence, the intersection of the left equivalence class relative
f
taining
-i
Let
{l,2,...,n}
D
S
be the set of all one-to-one functions
n
onto itself,
G
be a permutation group acting on
be a permutation group acting on the range
relation relative to
if there exist a
G
H
and
e G
and a
S
in
T
n
e H
D
be defined as follows:
-If (ad)
such that
and an equivalence
f
g(d)
if and only
g
for every
D
d
Then the number of patterns (equivalence classes) is
[PG(I
evaluated at
z
z
I
z
2
z2
__._
z 3 ,...) PH(Zl,2Z2,3z3,...)]
0.
3
(15) is also equal to
PG(Zl,2Z2,3z 3
evaluated at
Let
(12
p
p)
z
z
I
z
2
be a prime,
D
acting on
(15)
)]
(16)
0.
3
be the cyclic group
G
{l,2,...,p},
H
and
C
P
of order
generated by
p
D.
be the identity group acting on
By using (15) and Corollary 5.1 restated for one-to-one functions, it can be shown
that there are
order
C
P
p.
is
(p-l)!
right equivalence classes in
S
(p-l)!
C
relative to
P
Similarly, the number of left equivalence classes in
S
and the cardlnallty of each equivalence class is
P
each of
relative to
P
H
These results
p.
are in agreement with those concerning permutation polynomials over finite fields ob-
talned by Mullen in [6].
THEOREM 7.
q
Let
D
{1,2
is an integer greater than
(12...p)
acting on
D,
(12...q)
acting on
R.
and
i,
C
q
p}
and
C
be the cyclic group of order
P
R
{1,2,
q}
where
be the cyclic group of order
q
is a prime and
p
p
generated by
generated by
C.Y. CHAO AND C.I. DEISHER
758
(a)
If
q
is not a multiple of
C
relative to
C
and
If
q
(i.e.,
p
D
classes in
PROOF.
C
and
P
C
q
(a)
D
D
R
qiqT
Iwl
p-i
P
C
P
q
and
+ P
qP.
is,
q
),
then the number of strong equivalace
C
is
P
pP
(p-l)
I
P
pq,
2.
(qp-l-l)
T
does not involve
C
q
weak equivalence
and there is one weak equivalence class having
ip (x + (p-l)Xp)
q(zl+z2+’’’)
(P + (p-l)--) l(e
q
i
PCq (Xl’X2,...,xq)
IWI is
and
z. and
+ T)
z
P
has no fixed points and
]ZlfZ2ffi
(17)
.=0
The reason is that every non-
and
p
q
are relatively prime.
equal to
IW[
=
I
qp +
(p-l)q)
Applying our algorithm to the function
apl
C
and
P
i)and that there are
PCP (Xl,X2,...,Xp)
identity permutation in
a31
C
relative to
by (3), the number of weak equivalence classes
where the function
is
C
relative to
Since
(i)xqi li,
Hence, (17)
C
and
classes each having cardinality
=--I
u
R
We claim that the number of weak equivalence classes relative to
I
is _--
cardlnal+/-ty q.
D
R
consists of only one function, i.e., the number of
q
P
strong equivalence classes in
(b)
then every strong equivalence class in
p,
i,
I
fl
(qp-I
D
e R
+ P
(18)
I).
corresponding to
we have the weak equivalence class
fl
all- a21
consisting of the
q
functions corresponding to
all
a21
a31
apl
a12
a22
a32
ap2
a13
a23
a33
ap3
alq
a2q
a3q
apq
(19)
1.
Not counting the weak equivalence class above, we still have
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
P
(qp-I + P
qP
I)
1
-1 (qp-i
P
i)
Z59
Since there are
weak equivalence classes
functions and since each weak equivalence class can have its cardinallty at
1
the cardlnallty of each of these
pq,
most
P
(qp-1
1)
weak equlvalence classes
pq.
is
Now we show that every strong equivalence class in R
i’
C
relative to
C
and
P
q
Clearly, applying our algorlthm to each function in
only One function.
consists of
D
we have that each function belongs to a strong equivalence class" consisting of
Let
only itself.
u
i.e., there exist a
f
(.z-1)-lfe I
where
e
z
and a
P
and
e
C
q
fu
e2-1fo
But
C
C
and
P
q
D
First, we consider the set
claim that if
S
P
C
belongs to normalizer of
f
D
R
The cardinalfty of any right equivalence class in
S
u e C
e2-1fu
i.e.,
C
C
and
P
relative to
in the group
P
f
a
pq,
q
C
is i,
to
such that fu
P
is a normalizer of C
g.
Since
f
S
f
P
Sp,
C
and
D
C
exists.
C
is
P
C
in
S
P
P
belongs to
for every
u
C
Let
z
fuf
Since
C
P
in
f
S
P
fu
relative
then
C
and
P
Let
i.
is
in
C
P
f
f
and
such that
g
fu
fo
P
in S
and f is not normalizer of C
P
P
P
then the cardinality of the strong equivalence class containing f relative to C
We claim that if
p.
is
P
p.
Then there exists
P
Let
is a normalizer of
f
We
C
and
P
C
relative to
P
D.
(fuf-1)f.
We note that if
p.
is
P
P
is also a normalizer of
and
q
then the cardi-
relative to
be any two right equivalent functions relative to
g
C
C
and
p
Then Tf
C
T
and since u e C
P
P
P
Consequently, the cardinality of the strong equivalence class containing
f
q
f
Since
of all one-to-one functions in
nality of the strong equivalence class containing
and
respectlvely.
C
and
P
qP.
(b)
f
e2-1fu
C
relative to
relative to
i e., the number of the strong equivalence classes in
RDI
R
i’
Assume
g.
Tf
That is a contradiction, and the cardi-
nallty of every strong equlvalence class in R
is
D
has cardlnallty
f
(z-1)-lfe 1.
g
and
g
and not in
could be the identity, and we would have
z
the weak equlvalence class containing
(T-1)-lfel
such that
are the identities of
2
D
R
be strongly equivalent functions in
in the same weak equivalence class in
and
I
g
C
Then none of
g.
and
and
f
S
P
be strongly equivalent functions, i.e., there exist
g
and
f
g.
Assume that
f
g.
Then neither
C.Y. CHAO AND C.I. DEISHER
760
c
nor
fulf-i
(ff-l) (ff-l) ...(fuf-l)
ff
and
Tf
fc
is the identity,
rl
-I
T.
for
is not the identity,
Since
1,2,...,p,
i
f is a norm-
i.e.,
g, and the cardinality
That is a contradiction. Hence, f
P
and C
is i.
of the strong equivalence class containing f relative to C
P
P
D
then the cardlnality of the strong equiS
and f
We claim that if f e D
alizer of
C
S
in
P
p’
and C
P
P
in
classes
of weak equivalence
f
valence class containing
IWI
sider the number
relative to
C
I.
is
D
D
First, we shall con-
relative to
C
C
and
P
P
By using (3), we have
p(z +...)
z
P
pp-2
Let
..lpl
p
P
+ (p-l)e
P
]Zl--Z2=...=0
(2O)
+ p
i.
be the number of weak equivalence classes in
Sp
Cp
relative to
Cp.
and
By (15), we have
I ((p_l)! +
P
can only
Since a one-to-one function
(21)
(p-l)2).
be weakly equivalent to a one-to-one function and
since a non-one-to-one function can only be weakly equivalent to a non-one-to-one
function
lative to
II
the number
C
C
and
P
II
P
[WI
of weak equivalence non-one-to-one functions in
apl
I,
tlons corresponding to
D
re-
is
[g
P
(.pp-2
+ P
Applying our algorithm to the function
a31
D
I)
f2
1
e D
((p-l)! + (p-l) 2)
D
corresponding to
we have the weak equivalence class
f2
(22)
a
all
consisting of
p
2i
func-
EQUIVALENCE CLASSES OF FUNCTIONS ON FINITE SETS
all
a21
a31
apl
a12
a22
a32
ap2
761
i,
(23)
a
alp
a
a3p
2p
i.
pp
II
Not counting the weak equivalence class above, we still have
equivalence classes of non-one-to-one functions in
pP
p!
weak
DDI
Since there are
Sp
non-one-to-one functions and since each weak equivalence class can have
p2,
its cardinallty at most
valence classes is
P +
D.
D
i
p2[l_l
p
2
II
the cardlnality of each of these
I weak equl-
because
p2[(pp-2
P +
+ P
i)
Similar to the proof in (a) with
q
i
p,
(.(p-l)! + (p-l) 2]
p
pp
2
p!.
we may conclude that every strong
C
equivalence class of non-one-to-one functions relative to
C
and
P
consists of
P
only one function.
[DDI
We know that
C
in
S
is
pP,
p(p-l)
P
P
strong equivalence class
Sp[
p!
and the cardinallty of the normallzer of
(see 2.3 on p. 12 in [14]).
Since the cardlnallty of the
containing the normallzer
is also a normallzer of
and since every function in
strong equivalence classes each of which has cardinallty
C
of
f
C
P
p.
in
P
S
in
P
S
P
is
there are
p
p-i
Since every other
strong equivalence class has cardlnallty i, the number of strong equivalence classes
in
D
D
relative to
(pP
C
P
and
C
P
p!) + (p!
is
p(p-l)) + (p-l)
pP
(p-l)
2.
762
C.Y. CBAO AND C.I. DEISHER
REFACES
i.
Carlitz, L., Invarlatlve theory of equations in a finite field,
1953, 405-427.
.Math. S0c. 7__5,
2.
3.
T_ran. Ame r.
Cavlor, S.R., Equlvalence classes of functions over a finite fleld,
1__0, 1964, 119-136.
Acta Arlth.
Cavior, S.R.Equivalence classes of set of polynomlals over a finite fleld, Journ.
AnEew. Math. 25___5, (1967), 191-202.
Reine
4.
Mullen, G.L., Equlvalence classes of functions over a finite field, Acta Arith.
2_.9, (1976), 353-358.
5.
Mullen, G.L.Equlvalence classes of polynomials over finite lelds, Acta Arith.
31, (1975), 113-123.
6.
Mullen, G.L.Permutatlons polynomlal
Acta Arlth. 3__1, (1976), 107-111.
7.
Mullen, G..eak equlvalence of functions over a finite field, Acta Arith. 3__5,
(1979) 259-272.
8.
de BrulJn, N.G.,
9.
Harary, F. and Palmer, E., Graphlca!_Enumbera.tlon Academic Press, New York and
London, 1973.
10.
Plya, G., Kombinatorische Anzahlbestimmungen fur Gruppen Graphen und Chemische
Verbindungen, Acta Math. 68, (1937), 145-254.
ii.
Polya, G.
in several variables over finite fields,
Polya’ s theory of counting, Applied Combinatorlal Mathematics
(E.F. Beckenbach, ed.), Wiley, New York, 1964, 144-184.
Generalization of Plya’s fundamental theorem in enumerative combinatorial analysis, Nederl. Akad. Wetensch. Proc. Set. A 62
Ladagationes Mathe-
mathlcae 21, (1959), 59-69.
Enumeration of mapping patterns, J. Comb. Theory (A)
12.
Polya, G.
20.
13.
Chao, C.Y., On equlvalence classes of matrices, Bull. Malaysian Math. Soc. 4,
(1981), 29-36.
14.
Dixon, J.D., problems
in
Group
1__2, 1972, 14-
T..eory, B1aisde11, Waltham, Mass. 1967.