MacMahon extensively studied integer compositions, including the notion of conjugation. More recently, Agarwal introduced -color compositions and their cyclic versions were considered by Gibson, Gray, and Wang. In this paper, we develop and study a conjugation rule for cyclic -color compositions. Also, for fixed , we identify and enumerate the subset of self-conjugate compositions of , as well as establish a bijection between these and the set of cyclic regular compositions of with only odd parts.
A composition is an ordered list of positive
integers . Each of the numbers in the list is called a
part of size of the composition. The
sum of all the parts is called the weight of the
composition, and the number of parts (denoted ) is
called its length. We say is a
composition of (with
parts) if it has
weight and length . For any , we use to refer to the set of
regular compositions of ,
writing if we
also want to specify the number of parts.
Agarwal [1]
introduced the concept of -color
compositions. An -color
composition is a composition in which a part of size can be in one of different colors, which are usually
represented by numbered subscripts. For example, is a -color composition of with parts. We define and analogously to the
sets for regular compositions.
A well-known graphical representation of compositions is through
tilings. The scheme of
spotted tilings for -color compositions was introduced by
the first named author [8]. A part is represented by a tile with a spot marked on the
th square, as shown in Figure 1. This differs from
the spotless tiling that is used for regular compositions.
Figure 1 Tiling representation of the regular composition , and spotted tiling representation of the -color composition .
For both cases, we also have the notion of
cyclic compositions, in which we consider the
parts as being arranged around a circle, considering two arrangements
the same if one can be obtained by rotating the other. Regular cyclic
compositions were introduced by Sommerville [12] in 1909, while the -color case was treated by Gibson, Gray,
and the third named author [6] in 2018. If is a composition,
then the cyclic composition corresponding to it is denoted by .
The sets of regular and -color
cyclic compositions are denoted, respectively, by and
. The
graphical tilings can be easily extended to represent these cyclic
arrangements, as shown in Figure 2.
Figure 2 Tilings for the cyclic compositions and .
MacMahon [10] introduced conjugation for
regular compositions. We will present alternative representations of
-color compositions and cyclic
-color compositions that will
allow us to define conjugations on the cyclic version of these
compositions, and address related combinatorial questions. An equivalent
transformation was recently studied by Bowman [4] as a consequence of certain
bijections between different sets of compositions. Our approach builds
this map from the context of conjugation and helps to better understand
that transformation.
The spotted tilings will be helpful for visualizing connections
between regular and -color
compositions. In Section 2, we introduce three elementary transformations
on compositions and study the relationships between them. In Section 3, we
define the conjugation map and also count the number of self-conjugate
cyclic -color compositions. In
Section 4, we highlight two properties of the sequence
counting these compositions and establish a connection to certain
regular cyclic compositions. Section 5 briefly discusses
the limitations of expanding this conjugation of cyclic -color compositions to standard -color compositions. This work is
largely from the undergraduate honors thesis of the second named author
[2].
2. Preliminaries
2.1. A connection with regular compositions
We associate to an -color
composition a
length
regular composition which we call the expanded
form of the composition. These expanded forms can be
visualized through their corresponding tilings, as follows:
(a) Starting with an spotted tiling, replace every spotted cell with
a pair of consecutive spotted cells.
(b) Next, in each of these pairs, remove the spots and add a barrier
in the middle.
This results in the tiling of a valid regular composition. An example
is given in Figure 3. To reverse this
process, simply add a spot to the cells adjacent to every other barrier
(starting with the second one) and then merge each pair of adjacent
spotted cells into one, removing the barriers in the process.
Figure 3 The expanded form of is .
Similar representations have been studied in previous work: This a
slight modification from the -blocks and
tails introduced by the first and third named
authors [7], but here we do not
allow the tails to be empty. It is easy to see that the above map
introduces a bijection between these compositions, formally stated in
Lemma 2.1.
Lemma 2.1. There is a
one-to-one correspondence between the -color compositions of with parts and the regular compositions of
with parts.
Proof. The result is vacuously true when since both sets are empty. If
, for any -color composition , consider the transformation that turns each of the colored
parts (where ) into ordered pairs of
numbers . The
resulting list, obtained by concatenating each of the pairs (in the same
order), is a valid regular composition, because both and are positive integers. Furthermore,
the length of the new list is double that of the original. And since
, the weight of
the resulting composition will be .
The map is a
bijection because it has a straightforward inverse : Starting with a
composition , we generate an -color composition by turning each
successive pair of parts
into a colored part where .
It is natural to wonder about the expanded form of cyclic -color compositions. To answer this, we
study another transformation on compositions.
2.2. The shifts of a composition
If we define its shift,
as the composition . We use exponent notation
to denote repeated shifts. This map is invertible, so the notation makes
sense for any integer . For
example, given , we have The interaction between
and
is straightforward.
Because each part in the -color
composition corresponds to two parts in the expanded form, we have the
following.
Proposition 2.2. Let
be an -color composition. Then .
Proof. Let be the
last part of
and let be the
composition left after removing this last part from , i.e., and . The respective
expansions are thus .
Shifts provide a formal way to work with cyclic compositions. We
consider and
to be
related (written ) if there is an integer such that .
It is easy to verify that this is an equivalence relation and cyclic
compositions formally correspond to equivalence classes. In a similar
fashion, we say if there is an integer such that
(that is, if
can be obtained by shifting an even number of
times). Once again, this is an equivalence relationship. We denote the
corresponding equivalence classes by -cyclic compositions,
writing
for the -cyclic composition that
contains .
From Proposition 2.2 we deduce
that a cyclic -color composition
corresponds exactly to ,
a -cyclic composition of expanded
forms, which we call the cyclic expanded form of
.
Lemma 2.3. Let
be -color compositions. Then,
if and only if .
Proof. The assertion is true if and only if
for some integer , which in turn
occurs if and only if .
That is, if and only if .
Figure 4 The expanded form of a cyclic -color composition is a regular -cyclic composition.
Graphically, the cyclic expanded forms can be understood as cyclic
tilings in which there are two types of alternating barriers between the
parts, as shown on Figure 4. One of these
types corresponds to the usual barriers, but the second type arises from
the spotted squares.
2.3. On primitive compositions
In this section, we review properties of primitive compositions
studied by Lothaire [9]. Primitive compositions prove to
be a useful tool when studying cyclic compositions. These results are
used by Flajolet and Soria [5] to obtain generating
functions for general classes of cyclic compositions. Most of the
statements shown here apply to both regular and -color compositions.
We say a composition is primitive if it is not
the repeated self-concatenation of another composition. That is, is primitive if and
only if for some other composition and an integer . For example, the composition
is primitive, but the
composition is not. For every , there is exactly
one primitive composition such that , called the primitive
of , and
denoted by . In this case, we have . Notice that, while is a composition of
with parts, the list is a composition of
with parts. In general, we have the
following.
Proposition 2.4.
Let be compositions such that for some positive integer . The weight and length of are, respectively,
times the weight and length of
.
In particular, for any composition , the weight and
length of are divisors of the respective
quantities in . Now, notice that
for all we
have . For example, the
length of is , and indeed . We have is primitive if and
only if is
the smallest positive integer that satisfies this.
Proposition 2.5(Prop. 1.3.2, [9]). Let be an integer. A composition is primitive if and
only if implies .
How does the map sending compositions to their primitives interact
with the expanding and shifting transformations? First, if
are compositions, then if and only if for any integer we have In particular, we have the following.
Second, the expansion map (only defined for -color compositions) commutes with the
process of self-concatenation.
Proposition 2.7. Let
be -color compositions. We have if and only if .
Proof. The result is trivial when since then .
The rest of the proof follows by induction. Namely, if this is true for
a given , we must have if and only if
.
As a corollary, if an expanded form is
primitive, then the original composition must be primitive as
well.
3. A conjugation map
From the graphical depiction of the cyclic expanded forms, we deduce
a straightforward conjugation rule that works for all cyclic -color compositions. Namely, flipping
the roles of the alternating types of barriers produces the diagram of
another cyclic expanded form, which corresponds to a cyclic composition
of the same weight.
This can also be described directly from the spotted tilings. In
Figure 6, observe that the
number of barriers is exactly the same as the amount of marked
spots: There is exactly one marked spot between two consecutive
barriers, and vice versa. Thus, interchanging the marked spots and the
barriers will produce the diagram of another valid cyclic -color composition. Moreover, doing this
process twice returns the original composition. This interpretation was
concurrently developed by Bowman [4].
Figure 5 A conjugation on the expanded forms of cyclic $n$-color compositions extends to a conjugation on the cyclic compositions themselves.Figure 6 The conjugate of is the composition .
Formally, this is just the map that transforms cyclic -color compositions by expanding,
shifting (thus inverting the roles of all barriers), and contracting the
-color composition back into
another of the same weight. Recall that is the set of cyclic
-color compositions of weight
.
Theorem 3.1. The map is an involution on .
Proof. Notice that applying twice to an -color composition returns the -color composition , whose
expanded form is .
From here, , and thus as guaranteed by Lemma 2.3.
A natural question now is whether there are compositions for which
the conjugation acts as
the identity map. That is, whether there are
self-conjugate cyclic -color compositions. In this section, we
characterize and count such compositions of a fixed weight.
We begin by noting that, when fixing the weight, there is a
one-to-one correspondence between -cyclic compositions and their
primitives:
Proposition 3.2. Let
be two compositions of equal weight. Then, if and only if .
This follows from Proposition 2.6.
We are now able to characterize the set of self-conjugate cyclic
-color compositions.
Theorem 3.3. A cyclic -color composition is
self-conjugate if and only if has odd length.
Proof. Assume first that is
self-conjugate. That is, . From
the relationship between -color
compositions and their expanded forms given by Lemma 2.3, . In turn,
by Proposition 3.2. But by
Proposition 2.6, , so . Thus, there is an
integer such that
By Proposition 2.5, since is primitive by
definition. Since is odd, its
divisor must also be
odd.
For the other direction, assume is odd. We can
“split” the shift in two:
Because is even, . Equivalently, by
Proposition 2.6,
Finally, from Proposition 3.2,
and then Lemma 2.3 implies , i.e.,
is
self-conjugate.
For example, consider the cyclic -color composition . Note
that Since has odd
length, is self-conjugate by
Theorem 3.3; see Figure 7.
Figure 7 An example of a self-conjugate cyclic -color composition.
By the definition of primitive composition, any -color composition has for some integer . Applying Proposition 2.7 to and shows for
some integer . As powers of the
same word, .
We record these ideas in the following corollary.
Corollary 3.4. If
is
self-conjugate, then is also self-conjugate,
and so is any composition of the form , where is an integer.
Our next goal is enumerating self-conjugate cyclic -color compositions. It is actually
easier to count the complementary set, the compositions whose expanded
form has even length (i.e., those that are not self-conjugate).
We first analyze the case of primitive compositions, and then
extrapolate to non-primitive cases. In the next lemma we show that being
both primitive and self-conjugate determines the behavior of the
expanded form.
Lemma 3.5. Let be a primitive -color composition. The primitive of
is
either itself or
its exact half, that is, .
Moreover, the cyclic -color
composition is
self-conjugate if and only if it falls in the latter case.
Proof. By definition of the primitive composition, for
some integer . We need to show
is either or . By Proposition 2.4, the
length of is times that of . There are two cases
based on the parity of .
If is odd, since
is even, is also even.
But if the length of is even, then is the expanded form of
some other -color composition
. That is, for some other -color composition . By Proposition 2.7, undoing
the expansion yields which implies
because is
primitive.
If is even, say for some integer , then we can define so that . The length of is even, being
twice the length of . So, as above, there is
an -color composition such that . That is, and by, Proposition 2.7, , therefore which gives .
For the second claim, note that a primitive self-conjugate cyclic
-color composition
cannot have also
primitive, else the length of this primitive expanded form would be
even, contradicting Theorem 3.3. In other words, if
is
self-conjugate, then . We
saw above that having even length (that
is, not
being self-conjugate) means .
In other words, for primitive, is
not self-conjugate if and only if is also
primitive. We also know can only
be primitive when is also primitive.
Thus, counting the number of primitive non-self-conjugate cyclic -color compositions reduces to counting
the number of primitive expanded forms for a given weight . Write for
the number of primitive (regular) compositions of with parts.
Theorem 3.6. The
number of self-conjugate cyclic -color compositions of is
Proof. Recall that the expanded form of an -color composition of weight and length is a regular composition with weight
and length . For fixed , summing for possible values of (from to ) gives the number of primitive expanded
forms. That is, writing for
the number of primitive -color
compositions of whose expanded
form is also primitive,
Let be
the number of non-self-conjugate primitive cyclic -color compositions of a given weight
. Primitive expanded forms in the
same -cyclic composition
correspond with the same non-self-conjugate cycle, but by Proposition 2.5, a primitive
expanded form with length has
exactly even shifts. Therefore,
To complete the proof, we need an expression applicable to all
compositions, not just the primitive ones. Since fixing the weight causes a one-to-one relationship
between compositions and their primitives, we simply carry out an
additional sum of primitive compositions across all divisors of . Writing for the number
of non-self-conjugate cyclic -color compositions of a given weight
, we have
The total number of self-conjugate cyclic -color compositions of is then , as discussed earlier.
Table 1 gives the first few values of
which is now listed in the OEIS [11, A365859].
Table 1 Number of self-conjugate cyclic -color compositions of weight for .
4. On the sequence of self-conjugate cyclic -color compositions
The data of Table 1
suggest some general results: Through , it appears that .
Also, it seems that for all . We will prove that these results
hold in general.
First, we establish a consequence of Lemma 3.5.
Corollary 4.1. For even , there are no primitive
self-conjugate cyclic compositions of .
Proof. We show that for any primitive composition of , the corresponding cyclic
composition
cannot be self-conjugate. By Lemma 3.5
and Proposition 2.4, we have
where is either or . In any case, is an integer. Now, from
Lemma 2.1 we have . Thus, .
Lemma 2.1 also shows that the
length of is
. Because is
even, must
also be even. Therefore, is also even
since it is a multiple of .
Therefore,
cannot be self-conjugate.
With this, we can establish the patterns observed in Table 1.
Theorem 4.2. Given
a positive integer, .
Proof. We establish a bijection between the two sets of
self-conjugate compositions.
Let be
a self-conjugate cyclic -color
composition of . The
composition
is then a cyclic -color
composition of which is also
self-conjugate by Corollary 3.4. This
transformation is one-to-one because if and only if .
For the reverse map, let be
a self-conjugate cyclic -color
composition of . By
Corollary 4.1,
cannot be
primitive. However, is also self-conjugate by
Corollary 3.4 and it has an
odd weight which divides by Proposition 2.4. Any odd
divisor of is also a divisor
of . Thus, there is an integer
such that is a composition of
which, again by Corollary 3.4, is
self-conjugate. Notice that , so this mapping is the inverse
of the previous one.
The result for powers of 2 then follows from .
Another consequence of Theorem 4.2 is that the
subsequence of Table 1
from odd values determines the
entire sequence. This bisection is also in the OEIS [11,A365858] as the number of regular
cyclic compositions with only odd parts. We use expanded forms to
establish a bijection between these types of compositions.
Theorem 4.3. Let
be an odd integer, and . The number of self-conjugate
cyclic -color compositions of
with parts is the same as the number of
cyclic regular compositions of
with parts in which all parts are
odd.
Proof. Let be
a self-conjugate cyclic -color
composition of with parts. We know has parts and, since is
self-conjugate, has an odd length.
The length of must be an odd divisor
of the length of by
Proposition 2.4. As in
the proof of Theorem 4.2, this means
we halve , i.e.,
there exists a regular composition such that . Since has
weight and length , then must be a
composition of
with parts (note that, because
is odd, must also be odd). Now consider the
list obtained by
replacing each part of by . The weight of the resulting
composition is , as claimed.
This correspondence is invertible because, starting with , we can build the
composition ,
in which each part is changed
to , and then form an expanded
form
corresponding to an -color
composition . To see that is
self-conjugate, notice that . By Proposition 2.4, the
length of is a divisor of . But by parity, has an odd length since
it is a composition of an odd integer with only odd parts. Hence, the
length of is necessarily odd, which implies
that is
self-conjugate by Theorem 3.3.
We are left to show
if and only if , so that the map
sending to
is not only well-defined but also a bijection. By Lemma 2.3,
if and only if . We know
and
both have odd length. For compositions of odd length , even and odd shifts are equivalent: If
,
then . Thus, if and only if .
For example, consider again the cyclic -color composition of Figure 7 with
weight and parts which is also self-conjugate.
Disregarding the cyclic structure, we can split the expanded form into two
copies of . Taking
from each part in the second copy
leaves and . Adding these lists
element-wise gives ,
a regular composition of with
parts; see Figure 8.
Figure 8 Bijection between self-conjugate compositions and compositions with only odd parts.
5. Connection to balanced compositions
MacMahon defined conjugation for regular compositions and here we
have defined conjugation for cyclic -color compositions. The first named
author offered a notion of conjugation for -color compositions that unfortunately
only applies to a small subset [8]. The difference in the cyclic
case is that the number of barriers equals the number of spots as
described in Section 3. A definition of conjugation applicable to all
-color compositions remains
open.
In trying to address this problem, the second and third named authors
studied balanced-color compositions which themselves
have connections to many combinatorial objects [3]. The definition is based on
the expanded form: An -color
composition is
balanced if satisfies . For example,
is balanced
because its expanded form has .
We conclude with a connection between balanced -color compositions and the ideas
studied here.
Proposition 5.1. For ,
if is
self-conjugate, then is
balanced.
Proof. It suffices to show this for the primitive case. Let
. By Lemma 3.5,
and
is odd. By
Proposition 2.6, . Hence, for . Since is odd, this gives a natural matching
between the even and odd indexed parts of so that
is
balanced.
Certainly the converse does not hold, as the previous example is balanced but one can
verify that is
not self-conjugate.
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