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# Commuting pairs of functions on a finite set

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## Abstract

In this paper, we consider the problem of counting the number of ordered pairs (f, g) of commuting functions from [n] = {1, 2,. .. , n} to itself. We begin by considering the problem when the first function f is a permutation and derive an explicit formula in that case. As a consequence of our arguments, similar formulas may also be given when f belongs to one of several subsets of S n. An enumeration scheme and formula is developed for the problem of counting the ordered pairs of commuting functions with no restrictions. It relies on a classification of the functions from [n] to [n] based on how much a function differs from a permutation in some sense. Finally, it is shown that the number of ordered pairs of commuting functions from [n] to itself is always divisible by n.
Commuting pairs of functions on a ﬁnite set
Michael Holloway
Mathematics Department
University of Tennessee
Knoxville, TN 37996-1320
holloway@math.utk.edu
Mark Shattuck
Mathematics Department
University of Tennessee
Knoxville, TN 37996-1320
shattuck@math.utk.edu
Abstract
In this paper, we consider the problem of counting the number of ordered pairs
(f, g) of commuting functions from [n] = {1,2,...,n}to itself. We begin by consid-
ering the problem when the ﬁrst function fis a permutation and derive an explicit
formula in that case. As a consequence of our arguments, similar formulas may also
be given when fbelongs to one of several subsets of Sn. An enumeration scheme
and formula is developed for the problem of counting the ordered pairs of commut-
ing functions with no restrictions. It relies on a classiﬁcation of the functions from
[n] to [n] based on how much a function diﬀers from a permutation in some sense.
Finally, it is shown that the number of ordered pairs of commuting functions from
[n] to itself is always divisible by n.
1 Introduction
The problem of when the composition of two functions is commutative is a general one
that has been addressed in various settings. For example, in analysis, one such question
concerns the commutation of continuous functions from a closed interval to itself and
determining various properties of these functions such as common ﬁxed points (see [2]).
In linear algebra, the problem concerns the commutativity of square matrices of the
same size and one well-known result in this direction is that two real symmetric matrices
commute if and only if they are simultaneously diagonalizable (see, e.g., [5, p. 356]). Here
we consider the property of commutativity in a discrete setting and investigate various
aspects concerning the enumeration of pairs of commuting functions deﬁned on a ﬁnite
set.
Our results are also related to recent ones concerning the commutativity of pairs of
elements within ﬁnite semigroups (see [7, 1]). A theorem from algebra states that the
number of ordered pairs of commuting elements within a ﬁnite group Gis given by |G|
times the number of conjugacy classes of G(see, e.g., [6, p. 398]). When specialized to
the symmetric group Sn, this result states that the number of ordered pairs of commuting
permutations is given by n! times the number of partitions of n. See entry A053529 in
1
OEIS [8]. Various questions have been addressed in this direction; see, for example, the
paper by Erd˝os and Straus [4] where they consider maximal commuting k-tuples within
groups.
In this paper, we consider the problem of enumerating ordered pairs (f, g) of commut-
ing functions from the set [n] = {1,2,...,n}to itself. As a ﬁrst step, we consider the
problem of enumerating such pairs (f, g) where we require that fbe a permutation. In
the next section, we provide an explicit formula for the number of pairs in this case. Our
arguments also yield formulas for the number of commuting pairs (f, g) where fbelongs
to one of several subsets of Snand gis an arbitrary function. Using our formula, we can
also show that the number of commuting pairs (f, g) on [n] where fis a permutation is
always divisible by n.
The problem of determining an explicit formula or generating function for the number
of pairs of commuting functions from [n] to itself with no restriction though appears to
be more diﬃcult. Note that the sequence counting such pairs occurs as entry A181162 in
[8]. Here we provide a way of enumerating these pairs based on a certain partitioning of
the self-maps of [n]. As a consequence, we obtain a seemingly new combinatorial identity
relating nnto the multinomial coeﬃcients. Using the scheme we have developed, we are
able to prove that the number of ordered pairs of commuting functions from [n] to itself is
always divisible by n. We can also show that the probability of a randomly chosen pair of
elements commuting in the semigroup of self-maps on [n] converges to zero as nincreases
without bound.
We will make use of the following notational conventions. If mand nare positive
integers, then let [m, n] = {m, m + 1,...,n}if m6n, with [m, n] = if m > n. Empty
sums will assume the value zero and empty products the value one, with 00= 1. The
multinomial coeﬃcient, denoted n
n1,...,nr, is given by n!
n1!···nr!if n1,...,nrare non-negative
integers summing to nand is zero otherwise. Given a function fand a subset Sof its
codomain, the set of all elements xin the domain of fsuch that f(x)Swill be denoted
by f1(S).
2 Commuting functions and permutations
In this section, we consider the problem of counting the number of ordered pairs of
commuting functions from [n] to [n] where the ﬁrst function is a permutation. We begin
with the following lemma, which we’ll also need in the subsequent section.
Lemma 2.1. Given any function f: [n][n], there exists a non-empty subset Sof [n]
such that the restriction of fto Sis a permutation of S.
Proof. If i>0, then let fidenote the function fcomposed with itself itimes, where f0
is the identity function. Clearly, we have range(fi+1)range(fi) for all i. By ﬁniteness,
there exists some index jsuch that range(fj) = range(fj+1). Then the restriction of f
to range(fj) is clearly onto range(fj) and thus a permutation of the set since it is ﬁnite,
as desired.
2
Deﬁnition 2.2. We will refer to the restriction of fto range(fj)in the preceding proof
as the permutation base of fand to the set range(fj)itself as the base set of f.
Remark: Note that the base set of factually contains all subsets Ssuch that the restric-
tion of fto Sis a permutation of S(such subsets being contained within range(fi) for
all i).
Deﬁnition 2.3. Given a function f: [n][n], by a k-cycle of f, we will mean a
sequence (x1, x2,...,xk)of distinct entries in [n]such that xr=f(xr1)for 26r6k,
with x1=f(xk).
Lemma 2.4. Suppose f,gare functions from [n]to [n]. If fand gcommute, then for
any k-cycle (x1, x2,...,xk)of f, the sequence (g(x1), g(x2),...,g(x)) forms an -cycle of
ffor some |k.
Proof. Suppose fand gcommute. The fact that the images under gof a cycle of fform
a cycle follows from g(xi+1) = g(f(xi)) = f(g(xi)). If the length of the cycle doesn’t
divide k, then we have g(xk)6=g(x) and thus f(g(xk)) 6=f(g(x)) since g(x) and g(xk)
would be distinct elements belonging to the same cycle of f. But then g(x1) = f(g(x)) 6=
f(g(xk)) = g(f(xk)) = g(x1), a contradiction.
Remark: The converse of Lemma 2.4 also holds if fis a permutation.
Deﬁnition 2.5. Given a function f: [n][n], we will refer to the property possessed
by a commuting function gof mapping any k-cycle (x1, x2,...,xk)of fto an -cycle
(g(x1), g(x2),...,g(x)) of ffor some |kas the cycle mapping property.
Lemma 2.4 shows that the cycle structure of a permutation base of a function limits
which functions can commute with it. Given a permutation σ∈ Sn, we shall denote the
cycle structure of σby (λ1, λ2, . . . , λn), where λiis the number of cycles of σof length i.
Note that λ1+ 2λ2+···+n=n. We now revisit the problem of counting the number
of commuting pairs of permutations of [n] and provide a proof of the well-known formula
using the prior lemma.
Example. Recall that the number of commuting pairs of elements of Snis n! times the
number of partitions of n. To show this, note ﬁrst that by symmetry the number of
permutations of [n] commuting with a given permutation is the same for all permutations
possessing a given cycle structure. Let L=L(λ1,...,λn) denote the number of members
of Sncommuting with a permutation having a given cycle structure (λ1,...,λn). Then
the number of commuting pairs of permutations is given by
X
1λ1+···+n=n
λi>0
n!
λ1!···λn!1λ1···nλnL(λ1,...n),
by the well-known fact (see, e.g., [9, Prop. 1.3.2]) that there are n!
λ1!···λn!1λ1···nλnpermuta-
tions of length nhaving cycle structure (λ1,...,λn). To complete the proof, it suﬃces
3
to show that L=λ1!···λn!1λ1···nλn. Suppose σ∈ Snhas cycle structure (λ1, . . . , λn)
and that ρcommutes with σ. By Lemma 2.4 and injectivity, it is seen for each kthat ρ
permutes the k-cycles of σ. For each k-cycle of σ, there are kways to map it under ρto
any other k-cycle, whence there are kλkλk! ways to map k-cycles. Considering cycles of
all lengths implies L= 1λ1λ1!2λ2λ2!···nλnλn!, as desired.
Using a similar argument, we can obtain the following result.
Proposition 2.6. The number of commuting pairs of functions from [n]to [n]where the
ﬁrst function is a permutation is given by
X
1λ1+···+n=n
λi>0
n!
λ1!···λn!1λ1···nλn
n
Y
k=1
X
j|k
j
λk
.
Proof. For a permutation σof [n] with cycle structure (λ1, λ2,...,λn), the number of
functions g: [n][n] commuting with σis
n
Y
k=1
X
j|k
j
λk
.
Indeed, the image under gof each k-cycle of σis determined by choosing the image of a
single element in that cycle. By Lemma 2.4, this element may be mapped to any element
of any j-cycle where j|k. Summing over all possible cycle structures yields the result.
Apartition of a ﬁnite set is a collection of non-empty pairwise disjoint subsets, called
blocks, whose union is the set. A partition of [n] is said to have block structure (λ1, . . . , λn)
if it has λiblocks of size ifor each i. Recall that there are n!
λ1!···λn!1!λ1···n!λnpartitions of [n]
having block structure (λ1,...,λn). Note that partitions may be regarded as permutations
all of whose cycles are in ascending order. By restricting permutations to those corre-
sponding to set partitions in the proof of Proposition 2.6 above, we obtain the following
result.
Corollary 2.7. The number of commuting pairs of functions from [n]to [n]where the
ﬁrst function is a partition of [n]is given by
X
1λ1+···+n=n
λi>0
n!
λ1!···λn!1!λ1···n!λn
n
Y
k=1
X
j|k
j
λk
.
Remark. Note that the formula in Corollary 2.7 is that of the expression for the n-th
complete Bell polynomial Bn(t1, t2,...,tn) (see, e.g., [3]), but with the indeterminates tk
replaced by the sums Pj|kj.
4
Recall that an involution of [n] is a permutation all of whose cycles have length one or
two. By restricting the outer sum in the formula given in Proposition 2.6 above so that
only λ1or λ2may be non-zero, we obtain the following result.
Corollary 2.8. The number of commuting pairs of functions from [n]to [n]where the
ﬁrst function is an involution is given by
n!
n
X
k=0
kn(mod 2)
kk
k!nk
2!n
2nk
2.
Remark: By disallowing λ1>0 in the formula in Proposition 2.6, or by taking λ2=n
2
when nis even, one obtains expressions for the number of commuting pairs of functions
where the ﬁrst function is either a derangement (i.e., a permutation having no ﬁxed
points) or a perfect matching (i.e., a permutation all of whose cycles have length two).
Furthermore, taking λn= 1, it is seen that there are n! pairs of commuting functions,
where the ﬁrst function is a permutation containing a single cycle.
We have the following divisibility result concerning the number of commuting permu-
tation function pairs.
Theorem 2.9. The number of commuting pairs of functions from [n]to [n]where the
ﬁrst function is a permutation is divisible by nfor all n>1.
Proof. We make use of the formula given in Proposition 2.6. Let us group together all
terms for which is the smallest index such that λ>0, where 1 66nis ﬁxed. Let us
consider sums of this type. Note that the factors corresponding to 1 6k61 in the
product are all one, by 00= 1, and that the k=factor is given by (ℓλ)λsince λj= 0
if j < ℓ. Thus, the formula in this case may be written as
X
ℓλ+···+n=n
λ>0
n(n)!
ℓλ(λ1)!λ+1!···λn!λ1(+ 1)λ+1 ···nλn(ℓλ)λ
n
Y
k=+1
X
j|k,j>
j
λk
.
Since λ>0, by assumption, note that the ℓλfactor written at the beginning of the
denominator of the fraction is canceled out by (ℓλ)λ.
If 6n
2, then λ>0 implies λn=λn1=···=λn+1 = 0. In this case we may then
write
(n)!
(λ1)!λ+1!···λn!λ1(+ 1)λ+1 ···nλn
=(n)!
(λ1)!λ+1!···λn!λ1(+ 1)λ+1 ···(n)λn,
with the second expression seen to be an integer since it counts the number of permutations
of [n] having cycle structure (0,...,0, λ1, λ+1,...,λn), where it is understood
5
that there are 0’s in the ﬁrst 1 components of the vector. Since the product from
k=+ 1 to k=nin the sum above is clearly integral, it follows that the product of all
the factors is an integral multiple of n, and hence of n, since ℓ > 0.
If ℓ > n
2, then the minimality of implies =nand λ= 1, in which case we get a
single term of n!.
Thus the sum corresponding to each possible value is divisible by n. Considering all
possible , it follows that the total of all these sums is divisible by n, which implies the
result.
Remark: Using Corollary 2.8, one can show that the number of commuting pairs of
functions from [n] to [n] where the ﬁrst function is an involution is divisible by n2for all
n>3, with n3dividing the number of such pairs if and only if n= 2iand i>3.
3 Commuting pairs of functions
In this section, we address the problem of counting ordered pairs of commuting functions
deﬁned on the same ﬁnite set. Given a function f: [n][n], we consider the following
sequence of subsets of [n].
Deﬁnition 3.1. Given f: [n][n], deﬁne the sequence of sets (Bi)i>1by letting B1be
the base set of fand Bi+1 =f1(Bi)for i>1.
Lemma 3.2. If i>1and Bi6= [n], then BiBi+1.
Proof. We ﬁrst show by induction that BiBi+1 for i>1. Note that this holds if i= 1
since B1the base set of fimplies f(B1) = B1and thus B1f1(B1) = B2. If i>2, then
f(Bi)Bi1Biimplies Bif1(Bi) = Bi+1, which completes the induction. Now
suppose to the contrary that Bi=Bi+1 for some iwith Bi6= [n]. Then the restriction
of fto [n]Bihas range contained within [n]Bi. By Lemma 2.1, there exists some
non-empty subset Sof [n]Bisuch that frestricted to Sis a permutation of the set.
But this contradicts the earlier observation that B1contains all such subsets S, which
completes the proof.
We now consider the following way of classifying functions from [n] to [n].
Deﬁnition 3.3. We will say that f: [n][n]has rank rif Br= [n], where ris minimal.
Given a sequence (n1, n2,...,nr)of positive integers summing to n, we will say that fis
of type (n1, n2,...,nr)if ni=|Bi| − |Bi1|for i>1(where B0=).
Example. Let f: [10] [10] be given as follows:
f(1) = 1, f(2) = 1, f (3) = 1, f(4) = 3, f (5) = 2,
f(6) = 4, f(7) = 1, f (8) = 6, f(9) = 5, f (10) = 9.
6
Then B1={1},B2={1,2,3,7},B3={1,2,3,4,5,7},B4= [10]{8,10}, and B5= [10].
The function fwould have rank 5 and be of type (1,3,2,2,2). Note that the rank 1 func-
tions correspond to members of Snand are all of type (n).
Classifying the functions from [n] to [n] according to the rank and type yields the
following combinatorial identity which we were unable to ﬁnd in the literature.
Corollary 3.4. If n>1, then
nn=
n
X
r=1 X
Pr
i=1 ni=n
ni>0
n
n1, n2,...,nrn1!nn2
1nn3
2···nnr
r1.(3.1)
As we have seen, the permutation base of a function and its cycle structure limit which
functions can commute with it. On the other hand, any appropriate mapping of the cycles
of the permutation base of a function fcan be extended to the elements outside of the
base of fto produce a function which commutes with f. We start with a simple case.
Suppose fhas rank 2. Let us ﬁx a base of fand assume that it is a permutation of
[k] having cycle structure (λ1, . . . , λk). We enumerate the ordered pairs (f, g), where f
is of rank 2 and having a ﬁxed permutation base as described. Note that each member
of [k+ 1, n] maps to an element of [k] under f(knkpossibilities) since fis of rank 2
whose base set is [k]. To determine the functions commuting with fas described, we ﬁrst
choose a map α: [k][k] which possesses the cycle mapping property (so that there are
Qk
i=1 Pj|ijλipossibilities for α).
Now we need to determine how to map the elements of [k+ 1, n]. Let us denote the
restriction of fto [k+ 1, n] by β. If i[k+ 1, n] and β(i) = j, then a function g
commuting with fand extending αmay map ito any element which maps to α(j) under
f. Thus, there are |β1({α(β(i))})|+ 1 possibilities for the image of i. This implies that
there are
A(k;λ1,...,λk) := X
β:[k+1,n][k]
α:[k][k]
αpossesses c.m. property
Y
k+16i6n
(|β1({α(β(i))})|+ 1)
commuting pairs fand g, where fis of rank 2 and having a ﬁxed permutation base on
[k] with cycle structure (λ1,...,λk). Upon choosing the elements comprising the base set
of f, along with a permutation of those elements possessing a given cycle structure, it
follows that there are
n
X
k=1 n
kX
1λ1+···+k=k
λi>0
k!
λ1!···λk!1λ1···kλkA(k;λ1,...k)
commuting pairs fand g, where fhas rank 2.
7
In general, suppose fis a function of rank rand type (n1,...,nr). Assume that
Bi= [n1+···+ni] for i>1, where the Biare as deﬁned above with B0=. Then there
is a collection of functions f1:B1B1and fi:BiBi1Bi1Bi2for 2 6i6r,
where f1is a permutation, whose composite is f.
To form a function gcommuting with the function fwe have created, we need another
set of maps gigiving the images of each element under the commuting map. The following
lemma provides some structure for these maps.
Lemma 3.5. Let f: [n][n]have rank rand type (n1,...,nr). Suppose fis decomposed
into the functions fias described. If gcommutes with f, then gmay be decomposed into
functions g1:B1B1and gi:BiBi1Bifor 26i6r.
Proof. Suppose g(B1)Bi+1 for some 1 6i6r1. Then g(f(x)) = f(g(x)) for all
xB1implies g(B1)Bisince fpermutes the elements of B1and f(Bi+1)Bi. That
g(B1)B1now follows by induction since g(B1)Br= [n]. To complete the proof,
suppose to the contrary that g(BiBi1) is not contained within Bifor some i. Let i
denote the smallest such index i. By the preceding, we have i>1. Then there exists some
bBiBi1such that g(b)BjBj1where j > i. But then x=g(f(b)) = f(g(b))
implies xBi1(Bj1Bj2) = , a contradiction.
Note that the function g1in Lemma 3.5 possesses the cycle mapping property with
respect to the function f1, with f(gi(x)) = gi1(fi(x)) for all xBiBi1and i>2,
where the index , 1 66i, is determined by the set containing gi(x). The following
deﬁnition summarizes the properties discussed above that are possessed by commuting
pairs of functions.
Deﬁnition 3.6. Let Bi= [n1+n2+···+ni]for 16i6r, with B0=. Suppose {fi}r
i=1
is a collection of functions such that f1:B1B1is a permutation of type (λ1, λ2, . . . , λn1)
and fi:BiBi1Bi1Bi2for 26i6r. Let {gi}r
i=1 be a collection of functions
such that g1:B1B1commutes with f1and f(gi(x)) = gi1(fi(x)) for all xBiBi1
and i>2(the index ,166i, being determined by the set containing gi(x)). Then
we will denote the collection of all permissible sets of functions {fi}r
i=1 and {gi}r
i=1 by
S(n1,...,nr;λ1,...,λn1).
Allowing the elements in the Bito vary as well as the cycle structure of the permutation
base of fimplies the following result.
Proposition 3.7. The number of commuting pairs of functions from [n]to [n]is given
by n
X
r=1 X
n1+···+nr=n
ni>0
n
n1,...,nrX
1λ1+···+n1λn1=n1
λi>0
|S(n1,...,nr;λ1,...,λn1)|.
The following result concerns the probability that two elements in the semigroup of
self-maps on [n] commute (for related results, see [7]).
8
Theorem 3.8. The probability that a randomly chosen pair of functions from [n]to [n]
commutes converges to zero as n→ ∞.
Proof. Considering the size kof the base set of fin a commuting pair (f, g) of functions
on [n], it follows from Lemma 3.5 that the number of such pairs is bounded above by
Pn
k=1 n
kk!kkn2n2k. To complete the proof, it is enough to show
n
X
k=1
nkkk
n2k0 as n→ ∞.
Given n>1, let ak=nkkk
n2k, 1 6k6n. Using the fact that 1+k
kkis an increasing
function of khaving limit e < 3, along with the obvious inequality x(1 x)61
4for all x,
we have
ak+1
ak
=(nk)(k+ 1) 1+k
kk
n2<3
4+3(nk)
n2< c
for all k[n1] if nis suﬃciently large where 3
4< c < 1 is any ﬁxed constant. Thus,
n
X
k=1
nkkk
n2k<1
n
n
X
k=1
ck1<1
n(1 c)0 as n→ ∞,
as desired.
We conclude this section with the following divisibility result concerning the number
of ordered pairs of commuting functions.
Theorem 3.9. The number of commuting pairs of functions from [n]to [n]is divisible
by nfor all n>1.
Proof. For k= 1,...,n and λisuch that 1λ1+···+kλk=k, let C(λ1,...,λk) be the
number of commuting pairs of functions (f, g) such that the permutation base of fis
ﬁxed and has cycle structure (1λ1,...,kλk). Then the total number of commuting pairs
of functions from [n] to [n] is
n
X
k=1 n
kX
1λ1+···+k
λi>0
k!
λ1!···λk!1λ1···kλkC(λ1,...k).
As in the proof of Theorem 2.9 above, we group together all terms for which is the
smallest index such that λ>0, where 1 66kis ﬁxed. Then the formula in this case
is given by
n
X
k=1 X
ℓλ+···+k=k
λ>0
n
kk!
λ!···λk!λ···kλkC(0,...,0, λ,...,λk).
9
We show that this sum is divisible by nfor all kand ℓ. Since n
k=n
kn1
k1,it suﬃces to
show that k!
λ!···λk!λ···kλkC(0,...,0, λ,...,λk) is divisible by k. Writing
k!
λ!···λk!λ···kλk=k
ℓλ
·(k)!
(λ1)!λ+1!···λk!λ1(+ 1)λ+1 ···kλk,
and noting that the second factor is integral as in the proof of Theorem 2.9, to show that
k!
λ!···λk!λ···kλkC(0,...,0, λ,...,λk) is divisible by k, it suﬃces to show that ℓλdivides
C(0,...,0, λ,...,λk).
For a ﬁxed permutation base of the form (0,...,0, λ,...,λk),let C1,...,Cλbe the
cycles of length . Let P1be the set of ordered pairs (f, g) of commuting functions such
that fhas the given permutation base and f1(Ci) = Cifor some cycle Ci. We claim
that ℓλdivides |P1|. Let us assume that the index iis minimal and let mbe the smallest
element of Ci. By Lemma 2.4 and the minimality of , we see that g(m) must be some
member of [n] belonging to one of the cycles of length , which determines the functional
values of gfor all members of Ci. Thus we may partition the elements of P1into ℓλclasses
by requiring that g(m) be the same for all members of a given class. Since f1(Ci) = Ci,
it follows by symmetry that each class has the same cardinality, proving the claim.
Let P2be the set of ordered pairs (f, g) of commuting functions such that f1(Ci)6=Ci
for all i= 1,...,λ. Let D1=Sλ
i=1 Ci. For a function fwith the given permutation base,
let Df,1=D1and Df,i+1 =f1(Df,i)Df,i for i>1.We will deﬁne an equivalence relation
Ron P2by requiring that ordered pairs (f1, g1) and (f2, g2) be related if the following
three conditions are satisﬁed: (i) Df1,i =Df2,i for all i>1, (ii) f1|Si>3Df1,i =f2|Si>3Df2,i ,
and (iii) there exists a permutation σ: [n][n] such that σ(x) = xfor x /D1,
σ1f1σ=f2, and if a1,...,aD1, with a17→ a27→ · · · 7→ a7→ a1forming a cycle, then
σ(a1)7→ σ(a2)7→ · · · 7→ σ(a)7→ σ(a1) is a cycle.
Now ﬁx an equivalence class Eof R. Let Di=Df,i for any element (f, g) of E, and put
D=Si>1Di. We partition Eby (f1, g1)(f
1, g
1) if f1and f
1have the same restriction
to D2. Then there are λλ! equivalence classes under , since given any representative
(f, g) of a class of , the λλ! permutations σsatisfying condition (iii) above produce
representatives from each of the other equivalence classes by conjugation. Note that
(f, g)P2ensures that each σproduces a member belonging to a distinct equivalence
class of .
Let E1and E2be two equivalence classes of Eunder and suppose (f2, g2) is any
member of E2. Put f=f2|D. Then there is a unique σsatisfying condition (iii) above
such that σ1fσ =f1for any (f1, g1)E1.
Deﬁne φ:E1E2as follows. Given (f1, g1)E1, let f2(x) = f(x) for xD
and f2(x) = f1(x) for x /D. Note that σ1f2σ=f1.Now deﬁne g2=σg1σ1.Put
φ(f1, g1) = (f2, g2).Observe that we have
g2f2=σg1σ1f2=σg1f1σ1=σf1g1σ1=f2σg1σ1=f2g2.
The mapping φis seen to be a bijection, which implies all of the equivalence classes of E
under contain the same number of elements. Thus the size of each equivalence class
10
of Ris divisible by λλ!, which implies that |P2|is divisible by ℓλ. It follows that
C(0,...,0, λ,...,λk) is divisible by ℓλ, which completes the proof.
4 Conclusion
In this paper, we have considered the problem of enumerating ordered pairs (f, g) of com-
muting functions from [n] to itself. In the case where fbelongs to Sn(or one of several
subsets), an explicit formula counting these pairs is provided and a related divisibility
result was established. Building upon this case, an enumeration scheme and formula was
developed for the problem of counting ordered pairs of commuting functions with no re-
strictions. As a consequence, we can show that the number of ordered pairs of commuting
functions on [n] is always divisible by n. We seek though a more explicit formula and/or
generating function for the number of commuting pairs of functions, in particular for the
cardinalities of the sets S(n1, . . . , nr;λ1,...,λn1). It would also be interesting to have
asymptotic estimates as ngrows large for the number of such pairs. Finally, one might
consider the problem of enumerating ordered pairs of commuting elements in other ﬁnite
semigroup settings, such as the multiplication of n×nmatrices having integral entries
taken mod m.
References
[1] K. Ahmadidelir, C. M. Campbell, and H. Doostie, Almost commutative semigroups,
Algebra Colloq. 18 (2011) 881-888.
[2] H. Cohen, On ﬁxed points of commuting functions, Proc. Amer. Math. Soc. 15:2
(1964) 293–296.
[3] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht, Hol-
land, 1974.
[4] P. Erd˝os and E. G. Straus, How abelian is a ﬁnite group?, Linear Multilinear Algebra
3(4) (1975/76) 307–312.
[5] S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, Prentice Hall Inc.,
[6] J. A. Gallian, Contemporary Abstract Algebra, Houghton Miﬄin Company, Boston,
1998.
[7] B. Givens, The probability that two semigroup elements commute can be almost
anything, College Math. J. 39 (2008) 399–400.
[8] N. J. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2010.
[9] R. P. Stanley, Enumerative Combinatorics, Vol. I, Cambridge University Press, 1997.
11
2010 Mathematics Subject Classiﬁcation: Primary 05A05; Secondary 20M99.
Keywords: commuting functions, semigroup, commutativity.
12
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