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Jean-yves Thibon - One of the best experts on this subject based on the ideXlab platform.
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The Hopf algebras of signed Permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
Advances in Mathematics, 2020Co-Authors: Li Guo, Jean-yves ThibonAbstract:Abstract This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra S Sym of Permutations, mapped onto QSym by taking descents of Permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra H Sym , linearly spanned by signed Permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a Permutation from a signed Permutation defines a Hopf algebra surjection form H Sym to S Sym and taking a suitable descent from a signed Permutation defines a linear surjection from H Sym to RQSym. The notion of weak P-partitions from signed Permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from S Sym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, Permutations and signed Permutations.
Li Guo - One of the best experts on this subject based on the ideXlab platform.
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The Hopf algebras of signed Permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
Advances in Mathematics, 2020Co-Authors: Li Guo, Jean-yves ThibonAbstract:Abstract This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra S Sym of Permutations, mapped onto QSym by taking descents of Permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra H Sym , linearly spanned by signed Permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a Permutation from a signed Permutation defines a Hopf algebra surjection form H Sym to S Sym and taking a suitable descent from a signed Permutation defines a linear surjection from H Sym to RQSym. The notion of weak P-partitions from signed Permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from S Sym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, Permutations and signed Permutations.
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The Hopf algebras of signed Permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
2019Co-Authors: Li Guo, Thibon Jean-yves, Yu HouyiAbstract:This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of Permutations, mapped onto QSym by taking descents of Permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed Permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a Permutation from a signed Permutation defines a Hopf algebra surjection form HSym to SSym and taking a suitable descent from a signed Permutation defines a linear surjection from HSym to RQSym. The notion of signed $P$-partitions from signed Permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from SSym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, Permutations and signed Permutations.Comment: 25 pages, 2 figure
Philip B. Zhang - One of the best experts on this subject based on the ideXlab platform.
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On a Greedy Algorithm to Construct Universal Cycles for Permutations
International Journal of Foundations of Computer Science, 2019Co-Authors: Sergey Kitaev, Wolfgang Steiner, Philip B. ZhangAbstract:A universal cycle for Permutations of length [Formula: see text] is a cyclic word or Permutation, any factor of which is order-isomorphic to exactly one Permutation of length [Formula: see text], and containing all Permutations of length [Formula: see text] as factors. It is well known that universal cycles for Permutations of length [Formula: see text] exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for Permutations of length [Formula: see text], which is based on applying a greedy algorithm to a Permutation of length [Formula: see text]. We prove that this approach gives a unique universal cycle [Formula: see text] for Permutations, and we study properties of [Formula: see text].
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On a Greedy Algorithm to Construct Universal Cycles for Permutations
International Journal of Foundations of Computer Science, 2019Co-Authors: Alice L. L. Gao, Sergey Kitaev, Wolfgang Steiner, Philip B. ZhangAbstract:A universal cycle for Permutations of length $n$ is a cyclic word or Permutation, any factor of which is order-isomorphic to exactly one Permutation of length $n$, and containing all Permutations of length $n$ as factors. It is well known that universal cycles for Permutations of length $n$ exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for Permutations of length $n$, which is based on applying a greedy algorithm to a Permutation of length $n-1$. We prove that this approach gives a unique universal cycle $\Pi_n$ for Permutations, and we study properties of $\Pi_n$.
Sergey Kitaev - One of the best experts on this subject based on the ideXlab platform.
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On a Greedy Algorithm to Construct Universal Cycles for Permutations
International Journal of Foundations of Computer Science, 2019Co-Authors: Sergey Kitaev, Wolfgang Steiner, Philip B. ZhangAbstract:A universal cycle for Permutations of length [Formula: see text] is a cyclic word or Permutation, any factor of which is order-isomorphic to exactly one Permutation of length [Formula: see text], and containing all Permutations of length [Formula: see text] as factors. It is well known that universal cycles for Permutations of length [Formula: see text] exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for Permutations of length [Formula: see text], which is based on applying a greedy algorithm to a Permutation of length [Formula: see text]. We prove that this approach gives a unique universal cycle [Formula: see text] for Permutations, and we study properties of [Formula: see text].
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On a Greedy Algorithm to Construct Universal Cycles for Permutations
International Journal of Foundations of Computer Science, 2019Co-Authors: Alice L. L. Gao, Sergey Kitaev, Wolfgang Steiner, Philip B. ZhangAbstract:A universal cycle for Permutations of length $n$ is a cyclic word or Permutation, any factor of which is order-isomorphic to exactly one Permutation of length $n$, and containing all Permutations of length $n$ as factors. It is well known that universal cycles for Permutations of length $n$ exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for Permutations of length $n$, which is based on applying a greedy algorithm to a Permutation of length $n-1$. We prove that this approach gives a unique universal cycle $\Pi_n$ for Permutations, and we study properties of $\Pi_n$.
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On square-free Permutations
Journal of Automata Languages and Combinatorics, 2011Co-Authors: Sergey Avgustinovich, Sergey Kitaev, Artem V. Pyatkin, Alexandr ValyuzhenichAbstract:A Permutation is square-free if it does not contain two consecutive factors of length more than one that coincide in the reduced form (as patterns). We prove that the number of square-free Permutations of length n is nn(1"n) where "n ! 0 when n ! 1. A Permutation of length n is crucial with respect to squares if it avoids squares but any extension of it to the right, to a Permutation of length n+1, contains a square. A Permutation is maximal with respect to squares if both the Permutation and its reverse are crucial with respect to squares. We prove that there exist crucial Permutations with respect to squares of any length at least 7, and there exist maximal Permutations with respect to squares of odd lengths 8k+1; 8k+5; 8k+7 for k 1.
Yu Houyi - One of the best experts on this subject based on the ideXlab platform.
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The Hopf algebras of signed Permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
2019Co-Authors: Li Guo, Thibon Jean-yves, Yu HouyiAbstract:This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of Permutations, mapped onto QSym by taking descents of Permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed Permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a Permutation from a signed Permutation defines a Hopf algebra surjection form HSym to SSym and taking a suitable descent from a signed Permutation defines a linear surjection from HSym to RQSym. The notion of signed $P$-partitions from signed Permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from SSym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, Permutations and signed Permutations.Comment: 25 pages, 2 figure
