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Article

A Hopf Algebra on Permutations Arising from Super-Shuffle Product

School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2021, 13(6), 1010; https://doi.org/10.3390/sym13061010
Submission received: 7 May 2021 / Revised: 31 May 2021 / Accepted: 2 June 2021 / Published: 4 June 2021

Abstract

:
In this paper, we first prove that any atom of a permutation obtained by the super-shuffle product of two permutations can only consist of some complete atoms of the original two permutations. Then, we prove that the super-shuffle product and the cut-box coproduct on permutations are compatible, which makes it a bialgebra. As this algebra is graded and connected, it is a Hopf algebra.

1. Introduction

The concept of Hopf algebra was derived from the study of algebraic topology and algebraic groups in the 1940s and 1950s [1,2,3]. The structure of Hopf algebras is relatively stable and has many properties worth studying, such as commutativity, cocommutativity, and free and cofree properties [4,5,6]. With the further study of these properties, many new Hopf algebra structures have been discovered, especially in algebraic combinatorics, such as Hopf algebra on permutations [6,7,8], Hopf algebra on graphs [9], Hopf algebra on trees [6,10], and Hopf algebra on the symmetric functions [11,12]. In recent years, they have been gradually applied to many subjects, such as free Lie algebras [13,14], Lie superalgebras [15,16], Clifford algebras [17,18,19], Drinfeld-Jimbo quantum groups [20], the Jacobian Conjecture for free metabelian algebras [21,22] and even nonlinear control theory [23].
In 1995, Malvenuto and Reutenauer [24] studied a pair of Hopf algebra on permutations, where one of the products is essentially the shuffle product ш in [25]. The shuffle product’s ш is classic and has been studied by many teams recently. However, the product is not commutative on permutations. In 2020, Zhao and Li [26] introduced a new shuffle product ш G on permutations based on the global descents, which makes it commutative. Then, they defined a deconcatenation coproduct Δ and proved that ( K S , ш G , μ , Δ , ν ) is a graded connected bialgebra and, hence, a Hopf algebra [27]. Moreover, they figured out its dual Hopf algebra ( K S , ш G * , μ , Δ * , ν ) and a closed-formula of its antipode [28].
The symmetric group on a set X, denoted as S y m ( X ) , is a group of all bijections from X to itself, which is widely applied to many areas, such as algebraic number theory [29] and substochastic matrices [30,31,32,33]. In combinatorics, a permutation of degree n is an arrangement of n elements. The symmetric group of degree n, denoted by S n , contains all permutations of [ n ] = { 1 , 2 , , n } . Let K S n be the vector space spanned by S n over field K, where S n is the symmetric group of degree n. Define K S : = n 0 K S n , where S 0 = { ϵ } and ϵ is the empty permutation. Then, K S is graded, and its nth component is K S n .
In 2014, Vargas [34] defined the super-shuffle product ш ̲ on permutations, interpolating between the products ⋆ and given in [24], and a coproduct Δ , called the cut-box coproduct in our paper. He also put forward that the super-shuffle product ш ̲ is commutative and associative on permutations and claimed that ( K S , ш ̲ , μ , Δ , ν ) is a graded connected bialgebra and, hence, a Hopf algebra. However, he did not provide a proof.
In 2018, Bergeron, Ceballos, and Pilaud [35] gave a new concept of gaps on permutations in S n . In 2020, Aval, Bergeron, and Machacek [36] introduced global ascents of permutations and the standard factorization of a permutation given by global ascents. Then, the cut-box coproduct Δ of a permutation can be reinterpreted as using the tensor notation at one of its global ascents. In this paper, we prove that any atom of a permutation obtained by the super-shuffle product of two permutations can only consist of some complete atoms of the original two permutations, see Lemma 1. The main work of this paper is to prove that ( K S , ш ̲ , μ , Δ , ν ) is a bialgebra.
In Section 2, we recall some important definitions and notations on Hopf algebras and permutations. In Section 3, we introduce the definition of a super-shuffle product ш ̲ and reinterpret the cut-box coproduct Δ on permutations. We prove that ш and Δ satisfy the compatibility condition in Section 4. Therefore, ( K S , ш ̲ , μ , Δ , ν ) is a Hopf algebra since it is graded and connected. Finally, a summarization can be found in Section 5.

2. Preliminaries

2.1. Basic Definitions

Here, we recall some basic definitions related to Hopf algebra, see [3] for more details. Let R be a commutative ring and B be an R-module.
Define a product  π : B R B B and a unit μ : R B , respectively, satisfying the diagrams in Figure 1.
Then, ( B , π , μ ) is an R-algebra.
The algebra B is graded if there is a direct sum decomposition B = i 0 B i such that the product of homogeneous elements of degrees p and q is homogeneous of degree p + q , that is, π ( B p B q ) B p + q , and μ ( R ) B 0 .
Define a coproduct  Δ : B B R B and a counit ν : B R , respectively, satisfying the diagrams in Figure 2.
Then, ( B , Δ , ν ) is an R-coalgebra.
The coalgebra B is graded if there is a direct sum decomposition B = i 0 B i such that Δ ( B n ) ( B k B n k ) and ν ( B n ) = 0 if n 1 .
If B is both an R-algebra and an R-coalgebra, and satisfies one of the following equivalent conditions:
(1) Δ and ν are algebra homomorphisms;
(2) π and μ are coalgebra homomorphisms,
then we say the algebra and coalgebra structures on B are compatible and ( B , π , μ , Δ , ν ) is an R-bialgebra.
If B = i 0 B i is both a graded algebra and a graded coalgebra, and satisfies the compatibility condition, then we say B is a graded bialgebra.
If there exists a linear map θ : B B satisfying
π ( θ i d ) Δ = μ ν = π ( i d θ ) Δ ,
i.e., the diagram in Figure 3 commutes, then θ is an antipode. A bialgebra with an antipode is a Hopf algebra.
A bialgebra B over K is called graded connected if it is graded and satisfies B 0 = K 1 B , where K is a field. In 2008, Manchon [27] proved that any graded connected bialgebra admits a unique antipode and is a Hopf algebra.

2.2. Basic Notations

Now, we recall some basic notations on permutations; see [34,36] for more details.
Let S n be the symmetric group of degree n, which consists of all the permutations of degree n. A permutation ω in S n is a rearrangement of [ n ] : = { 1 , 2 , , n } , denoted by ω = ω 1 ω i ω i + 1 ω n in one-line notation, where ω i = ω ( i ) [ n ] . Let S : = n 0 S n be the disjoint union of S n , where S 0 = { ϵ } and ϵ is the empty permutation.
For a permutation ω = ω 1 ω 2 ω n in S n in one-line notation, the position in front of the first element ω 1 is called gap 0 of ω , the position between ω i and ω i + 1 is called gapi of ω for all 0 < i < n , and the position behind the last element ω n is called gap n of ω . For example, the gaps of permutation 321465 S 6 are the numbers in the circles: 3 2 1 4 6 5 .
For a gap γ of ω , if ω ( [ γ ] ) = [ γ ] , then γ is called a global ascent of ω . The gaps 0 and n must be global ascents of any permutation of degree n. A nonempty permutation with no global ascents except 0 and n is called an indecomposable permutation or an atom. A portion of an atom is a nontrivial subsequence of k adjacent elements from an atom with m elements, where 0 < k < m . For example, permutations 321 , 312 , 21 , and 1 are all atoms. The portions of 321 can only be 3 , 2 , 1 , 32 , and 21.
A sequence of [ n ] with length m is denoted by a = a 1 a 2 a m ( 1 m n ) , where a i ( 1 i m ) are finitely distinct elements selected from [ n ] . When no element is selected, we have an empty sequence with length 0, also denoted by ϵ . Let alph ( a ) be the set consisting of all elements in a and A m be the set of all the sequences of length m. The standard form of sequence a in A m is a permutation in S m , denoted by st ( a ) , satisfying
st ( a i ) < st ( a j ) a i < a j ,
st ( a i ) > st ( a j ) a i > a j .
For example, the standard form of sequence 432 is 321, i.e., st ( 432 ) = 321 . The standard form of the empty sequence is the empty permutation, i.e., st ( ϵ ) = ϵ .
Consider a permutation α S n . By putting boxes ⋄ at all global ascents except 0 and n, we find α = α 1 α l , called the factorization of α , where α i ’s are sequences of [ n ] and the sum of the length of all α i ’s is n. For example, let α = 2314657 , then its global ascents are 0 , 3 , 4 , 6 , 7 and its factorization is α = 231 4 65 7 , where α 1 = 231 , α 2 = 4 , α 3 = 65 , α 4 = 7 .
By taking the standard form of each sequence α i ( 1 i l ) , we obtain α ’s standard factorization α = st ( α 1 ) st ( α l ) . Each permutation has a unique standard factorization. Clearly, each st ( α i ) ( 1 i l ) in the standard factorization of α is an atom. The number of atoms in α is called the length of α , denoted as | α | , so | α | = l , which is always less than or equal to its degree n. From now on, the notation α = α 1 α l always expresses α ’s standard factorization. For example, the standard factorization of α = 2314657 is α = st ( 231 ) st ( 4 ) st ( 65 ) st ( 7 ) = 231 1 21 1 , where α 1 = 231 , α 2 = 1 , α 3 = 21 , α 4 = 1 , and the length of α is | α | = | 231 1 21 1 | = 4 .
From a standard factorization to the original permutation, we renumber the numbers in the atoms from left to right. For example,
21 12 1 1 = 213456 ,
1 12 12 1 = 123456 .

3. Super-Shuffle Product and Cut-Box Coproduct

In 2014, Vargas defined two operations on permutations—the super-shuffle product and cut-box coproduct as follows [34].
Define the cut-box coproduct  Δ on K S by
Δ ( α ) : = i = 0 l α 1 α i α i + 1 α l ,
for any permutation α = α 1 α 2 α l .
Example 1.
For permutation 214356, its global ascents are 0 , 2 , 4 , 5 , 6 , and its standard factorization is 21 21 1 1 . Then,
Δ ( 214356 ) = Δ ( 21 21 1 1 ) = ϵ 21 21 1 1 + 21 21 1 1 + 21 21 1 1 + 21 21 1 1 + 21 21 1 1 ϵ = ϵ 214356 + 21 2134 + 2143 12 + 21435 1 + 214356 ϵ .
From above, in the coproduct of a permutation the tensor notation ⊗ can only appear in one of its global ascents.
The counit map on K S is defined by ν : K S K , ν ( ϵ ) = 1 and ν ( α ) = 0 for any α S n , n 1 .
Let I , J [ n ] , and [ n ] be the disjoint union of I and J, denoted by I J = [ n ] .
For a permutation ω = ω 1 ω n S n , let ω I = ω i 1 ω i k with I = { i 1 < < i k } [ n ] .
Define super-shuffle product on K S by
α ш ̲ β : = I J = [ n + p ] s t ( ω I ) = α , s t ( ω J ) = β a l p h a ( ω I ) a l p h a ( ω J ) = [ n + p ] ω
for any α S n , β S p .
Example 2.
Let α = 12 S 2 , β = 1 S 1 .
12 ш ̲ 1 = 12 3 + 1 3 2 + 3 12 + 13 2 + 1 2 3 + 2 13 + 23 1 + 2 1 3 + 1 23 .
Here we color the elements of α red and the elements of β blue, then the same numbers in α and β are different. Thus, we can claim that all permutations in the super-shuffle product are distinct even if they are same. From now on, when we compute super-shuffle product of two permutations we will color them in different colors.
For α = α 1 α q S q , define α ^ i ( 1 i q ) to be the renumbering of α i satisfying that, if r t and α r < α t , then α ^ r < α ^ t , for all r , t [ q ] .
Remark 1.
For α = α 1 α q S q , β = β 1 β p S p . If T : = { α ^ i | 1 i q } [ q + p ] and { β ^ j | 1 j p } = [ q + p ] \ T , then
π : = α ^ 1 α ^ r β ^ 1 α ^ t β ^ k α ^ q β ^ s β ^ p
is a permutation in the super-shuffle product α ш ̲ β , where 0 r q , r + 1 t q , 1 k s 1 , 1 s p + 1 . In particular, when r = 0 the first element of π is β ^ 1 , and when s = p + 1 , the last element of π is α ^ q .
Therefore, 1 ^ 2 ^ 1 ^ is one of permutations in { 12 3 , 13 2 , 23 1 } , 1 ^ 1 ^ 2 ^ is one of permutations in { 1 3 2 , 1 2 3 , 2 1 3 } , and 1 ^ 1 ^ 2 ^ is one of permutations in { 3 12 , 2 13 , 1 23 } .
Define the unit map μ : K K S on K S by μ ( 1 ) = ϵ .
Define the relative shuffle product  ш on K S recursively by
α ш β = α 1 ( α 2 α l ш β ) + β 1 ( α ш β 2 β k ) ,
for α = α 1 α l , β = β 1 β k and α ш ϵ = ϵ ш α = α .
For example,
( 21 1 ) ш 1 = 21 1 1 + 21 1 1 + 1 21 1 = 213 4 + 21 3 4 + 1 324 .
For α S n and β S p , the permutations obtained by α ш ̲ β of degree n + p , which are the permutations obtained by α ш β and some others with length less than | α | + | β | .
For example, let α = 12 S 2 , β = 1 S 1 , and then | α | = | 1 1 | = 2 , | β | = 1 , | α | + | β | = 3 . Then,
12 ш 1 = ( 1 1 ) ш 1 = 1 1 1 + 1 1 1 + 1 1 1 = 12 3 + 1 2 3 + 1 23 ,
12 ш ̲ 1 = 12 3 + 1 2 3 + 1 23 12 ш 1 + 13 2 + 1 3 2 + 2 1 3 + 2 13 | π | = 2 + 23 1 + 3 12 | π | = 1 .

4. The Hopf Algebra Arising from the Super-Shuffle

By the definition, we have that the super-shuffle product of any two permutations of degrees n and p is a sum of permutations of degree n + p . Therefore, ( K S , ш ̲ , μ ) is a graded algebra. Since Δ ( K S n ) k = 0 n ( K S k K S n k ) , ν ( K ) = K S 0 , ( K S , Δ , ν ) is a graded coalgebra.
Vargas [34] put forward that ( K S , ш ̲ , μ , Δ , ν ) is a graded bialgebra, and thus it is a graded Hopf algebra. However, there was no proof. Here, we give a proof.
Lemma 1.
Let σ = σ 1 σ s , ρ = ρ 1 ρ t be any two nonempty permutations. If π is a permutation obtained by σ ш ̲ ρ , then any atom of π can only consist of some complete atoms of σ and ρ.
Proof 
(Proof of Lemma 1). Equivalently, we will prove that any atom of π does not contain any portion of atoms of σ or ρ .
Let π = π 1 π 2 π s + t in one-line notation be a permutation obtained by σ ш ̲ ρ and st ( π u π v ) is an atom of π , where 1 u v s + t .
Suppose π u π v contains a portion of an atom from the original permutation σ . Let st ( σ 1 σ p ) be the portion, where σ p σ q is corresponding to the portion in the one-line notation of σ . Then neither σ ^ p 1 nor σ ^ q + 1 is an element in π u π v .
Since π u π v is an atom, we have π ( [ u 1 ] ) = [ u 1 ] and π ( [ v ] ) = [ v ] . So, { σ ^ 1 , , σ ^ p 1 } [ u 1 ] , { σ ^ p , , σ ^ s } [ s + t ] \ [ u 1 ] , { σ ^ 1 , , σ ^ q } [ v ] and { σ ^ q + 1 , , σ ^ s }
[ s + t ] \ [ v ] . By the definition of renumbering, max { σ 1 , , σ p 1 } < min { σ p , , σ s } , max { σ 1 , , σ q } < min { σ q + 1 , , σ s } and { σ 1 , , σ s } = [ n ] . Thus, σ ( [ p 1 ] ) = [ p 1 ] and σ ( [ q ] ) = [ q ] . Clearly, σ ( [ k ] ) [ k ] for all p < k < q . Therefore, p 1 and q are global ascents of σ , and there is no global ascent between p and q 1 , i.e., st ( σ p σ q ) is an atom in σ , contradiction to that st ( σ p σ q ) is a portion.
Similarly, π u π v cannot contain a portion of any atom from the original permutation ρ . □
Example 3.
For any permutation α = 21 and β = 1 1 ,
21 ш ̲ 1 1 = 43 12 + 4 1 3 2 + 4 12 3 + 1 43 2 + 1 4 2 3 + 1 2 43 + 42 13 + 4 1 2 3 + 4 13 2 + 1 42 3 + 1 4 3 2 + 1 3 42 + 41 23 + 4 2 1 3 + 4 23 1 + 2 41 3 + 2 4 3 1 + 23 41 + 32 1 4 + 3 1 2 4 + 3 14 2 + 1 32 4 + 1 3 4 2 + 1 4 32 + 31 2 4 + 3 2 1 4 + 3 24 1 + 2 31 4 + 2 3 4 1 + 24 31 + 21 3 4 + 2 3 1 4 + 2 34 1 + 3 21 4 + 3 2 4 1 + 34 21 .
In our example, the atoms of each permutation obtained by 21 ш ̲ 1 1 all consist of some complete atoms of α and β .
Theorem 1.
( K S , ш ̲ , μ , Δ , ν ) is a bialgebra.
Proof of Theorem 1.
We know that ( K S , ш ̲ , μ , Δ , ν ) is a graded algebra and a graded coalgebrae. It is easy to check that ν is an algebra homomorphism. Thus, to prove ( K S , ш ̲ , μ , Δ , ν ) is a graded bialgebra we only need to prove it satisfies the compatibility, i.e.,
Δ ( α ш ̲ β ) = Δ ( α ) ш ̲ Δ ( β ) ,
for any α = α 1 α l and β = β 1 β t .
From Lemma 1, we have that any atom of a permutation π obtained by α ш ̲ β can only consist of some complete atoms from the original two permutations α and β .
Denote Δ ( α ш ̲ β ) | i = x as the sum of all items in Δ ( α ш ̲ β ) with x atoms from α on the left side of tensor notation ⊗, 0 x l . Hence, we have
Δ ( α ш ̲ β ) = Δ ( α ш ̲ β ) | i = 0 + Δ ( α ш ̲ β ) | i = 1 + Δ ( α ш ̲ β ) | i = 2 + + Δ ( α ш ̲ β ) | i = l ,
where
Δ ( α ш ̲ β ) | i = x = α 1 α x ( α x + 1 α l ш ̲ β ) + ( α 1 α x ш ̲ β 1 ) ( α x + 1 α l ш ̲ β 2 β t ) + ( α 1 α x ш ̲ β 1 β 2 ) ( α x + 1 α l ш ̲ β 3 β t ) + + ( α 1 α x ш ̲ β ) ( α x + 1 α l ш ̲ ϵ )
= j = 0 t ( α 1 α x ш ̲ β 1 β j ) ( α x + 1 α l ш ̲ β j + 1 β t ) = ( α 1 α x α x + 1 α l ) ш ̲ j = 0 t ( β 1 β j β j + 1 β t ) .
Thus, we have
Δ ( α ш ̲ β ) = Δ ( α ш ̲ β ) | i = 0 + Δ ( α ш ̲ β ) | i = 1 + Δ ( α ш ̲ β ) | i = 2 + + Δ ( α ш ̲ β ) | i = l = ( ϵ α 1 α l ) ш ̲ j = 0 t ( β 1 β j β j + 1 β t ) + ( α 1 α 2 α l ) ш ̲ j = 0 t ( β 1 β j β j + 1 β t ) + ( α 1 α 2 α 3 α l ) ш ̲ j = 0 t ( β 1 β j β j + 1 β t ) + + ( α 1 α l ϵ ) ш ̲ j = 0 t ( β 1 β j β j + 1 β t ) = ϵ α 1 α l + α 1 α 2 α l + + α 1 α l ϵ ш ̲ j = 0 t ( β 1 β j β j + 1 β t ) = i = 0 l α 1 α i α i + 1 α l ш ̲ j = 0 t ( β 1 β j β j + 1 β t ) = Δ ( α ) ш ̲ Δ ( β ) .
Hence, the cut-box coproduct Δ is an algebra homomorphism, i.e., the super-shuffle product and the cut-box coproduct are compatible. Therefore, ( K S , ш ̲ , μ , Δ , ν ) is a bialgebra. □
Example 4.
Let α = 12 = 1 1 , β = 1 . Then,
Δ ( 12 ш ̲ 1 ) = Δ ( 23 1 + 2 1 3 + 1 2 3 + 1 3 2 + 1 2 3 + 2 1 3 + 1 2 3 + 1 3 2 + 3 12 ) = Δ 23 1 + 2 1 1 + 1 1 1 + 1 2 1 + 1 1 1 + 2 1 1 + 1 1 1 + 1 2 1 + 3 12 = ϵ 23 1 + 2 1 1 + 1 1 1 + 1 2 1 + 1 1 1 + 2 1 1 + 1 1 1 + 1 2 1 + 3 12 + 1 1 1 + 1 ( 2 1 + 1 1 + 2 1 ) + ( 2 1 + 1 1 + 2 1 ) 1 + 1 1 1 + 23 1 + 2 1 1 + 1 1 1 + 1 2 1 + 1 1 1 + 2 1 1 + 1 1 1 + 1 2 1 + 3 12 ϵ = ϵ ( 1 1 ш ̲ 1 ) + 1 ( 1 1 ) + 1 ( 1 ш ̲ 1 ) + ( 1 ш ̲ 1 ) 1 + 1 1 1 + ( 1 1 ш ̲ 1 ) ϵ ,
and
Δ ( 12 ) ш ̲ Δ ( 1 ) = ( ϵ 1 1 ) ш ̲ ( ϵ 1 ) + ( 1 1 ) ш ̲ ( ϵ 1 ) + ( 1 1 ϵ ) ш ̲ ( ϵ 1 ) + ( ϵ 1 1 ) ш ̲ ( 1 ϵ ) + ( 1 1 ) ш ̲ ( 1 ϵ ) + ( 1 1 ϵ ) ш ̲ ( 1 ϵ )
= ( ϵ ш ̲ ϵ ) ( 1 1 ш ̲ 1 ) + ( 1 ш ̲ ϵ ) ( 1 ш ̲ 1 ) + ( 1 1 ш ̲ ϵ ) ( ϵ ш ̲ 1 ) + ( ϵ ш ̲ 1 ) ( 1 1 ш ̲ ϵ ) + ( 1 ш ̲ 1 ) ( 1 ш ̲ ϵ ) + ( 1 1 ш ̲ 1 ) ( ϵ ш ̲ ϵ ) .
Thus, we have Δ ( 12 ш ̲ 1 ) = Δ ( 12 ) ш ̲ Δ ( 1 ) .
Corollary 1.
( K S , ш ̲ , μ , Δ , ν ) is a Hopf algebra.
Proof of Corollary 1.
From Theorem 1, we know ( K S , ш ̲ , μ , Δ , ν ) is a graded bialgebra. Since ( K S , ш ̲ , μ , Δ , ν ) is graded and connected, ( K S , ш ̲ , μ , Δ , ν ) is a Hopf algebra. □

5. Conclusions

First, we proved that any atom of a permutation π obtained by α ш ̲ β can only consist of some complete atoms of α and β . Secondly, we proved that the super-shuffle product ш ̲ and the cut-box coproduct Δ on K S are compatible, and hence ( K S , ш ̲ , μ , Δ , ν ) is a bialgebra; Finally, since this bialgebra is graded and connected, it is also a Hopf algebra.

Author Contributions

Conceptualization, M.L. and H.L.; methodology, H.L.; investigation, M.L.; writing—original draft preparation, M.L.; writing—review and editing, M.L. and H.L.; supervision, H.L.; project administration, H.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by National Natural Science Foundation of China (Nos. 11701339, 12071265 and 11771256).

Data Availability Statement

No real data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Associative law and unit.
Figure 1. Associative law and unit.
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Figure 2. Coassociative Law and Counit.
Figure 2. Coassociative Law and Counit.
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Figure 3. Antipode.
Figure 3. Antipode.
Symmetry 13 01010 g003
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Liu, M.; Li, H. A Hopf Algebra on Permutations Arising from Super-Shuffle Product. Symmetry 2021, 13, 1010. https://doi.org/10.3390/sym13061010

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Liu M, Li H. A Hopf Algebra on Permutations Arising from Super-Shuffle Product. Symmetry. 2021; 13(6):1010. https://doi.org/10.3390/sym13061010

Chicago/Turabian Style

Liu, Mengyu, and Huilan Li. 2021. "A Hopf Algebra on Permutations Arising from Super-Shuffle Product" Symmetry 13, no. 6: 1010. https://doi.org/10.3390/sym13061010

APA Style

Liu, M., & Li, H. (2021). A Hopf Algebra on Permutations Arising from Super-Shuffle Product. Symmetry, 13(6), 1010. https://doi.org/10.3390/sym13061010

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