Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang Department of Mathematics and Computer Science Davidson College July 10, 2018 16th International Conference on Permutation Patterns Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Partially based on joint work with Ira M. Gessel. Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Outline 1 Introduction 2 Three Basic Homomorphisms 3 Homomorphisms Arising from Shuffle-Compatibility Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Outline 1 Introduction 2 Three Basic Homomorphisms 3 Homomorphisms Arising from Shuffle-Compatibility Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Permutations and Descents • A permutation of length n (or, n -permutation) π = π 1 π 2 · · · π n is a linear ordering of [ n ] := { 1 , 2 , . . . , n } . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Permutations and Descents • A permutation of length n (or, n -permutation) π = π 1 π 2 · · · π n is a linear ordering of [ n ] := { 1 , 2 , . . . , n } . • The set of n -permutations is denoted S n . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Permutations and Descents • A permutation of length n (or, n -permutation) π = π 1 π 2 · · · π n is a linear ordering of [ n ] := { 1 , 2 , . . . , n } . • The set of n -permutations is denoted S n . • We say that k ∈ [ n − 1 ] is a descent of π ∈ S n if π k > π k + 1 . The descent number des( π ) is the number of descents of π . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Increasing Runs and Descent Compositions • Descents separate permutations into increasing runs: maximal increasing consecutive subsequences. • Call the tuple of increasing run lengths of π the descent composition of π , denoted Comp( π ) . Example Given π = 172346589, we have Comp( π ) = ( 2 , 4 , 3 ) . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Increasing Runs and Descent Compositions • Descents separate permutations into increasing runs: maximal increasing consecutive subsequences. • Call the tuple of increasing run lengths of π the descent composition of π , denoted Comp( π ) . Example Given π = 172346589, we have Comp( π ) = ( 2 , 4 , 3 ) . • The notions of descents and increasing runs extend to words on any totally ordered alphabet (such as the positive integers P ). Example The increasing runs of the word w = 11526249 are 115, 26, and 249, so Comp( w ) = ( 3 , 2 , 3 ) and des( w ) = 2. Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Ribbon Functions • Let X 1 , X 2 , . . . be noncommuting variables. Given a composition L = ( L 1 , L 2 , . . . , L k ) , define the ribbon function r L by � r L := X i 1 X i 2 · · · X i n ( i 1 ,..., i n ) over all ( i 1 , . . . , i n ) satisfying i 1 ≤ · · · ≤ i L 1 > i L 1 + 1 ≤ · · · ≤ i L 1 + L 2 > · · · > i L 1 + ··· + L k − 1 + 1 ≤ · · · ≤ i n . � �� � � �� � � �� � L 1 L 2 L k Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Ribbon Functions • Let X 1 , X 2 , . . . be noncommuting variables. Given a composition L = ( L 1 , L 2 , . . . , L k ) , define the ribbon function r L by � r L := X i 1 X i 2 · · · X i n ( i 1 ,..., i n ) over all ( i 1 , . . . , i n ) satisfying i 1 ≤ · · · ≤ i L 1 > i L 1 + 1 ≤ · · · ≤ i L 1 + L 2 > · · · > i L 1 + ··· + L k − 1 + 1 ≤ · · · ≤ i n . � �� � � �� � � �� � L 1 L 2 L k • r L is the noncommutative generating function for words on the alphabet P with descent composition L . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Ribbon Functions • Let X 1 , X 2 , . . . be noncommuting variables. Given a composition L = ( L 1 , L 2 , . . . , L k ) , define the ribbon function r L by � r L := X i 1 X i 2 · · · X i n ( i 1 ,..., i n ) over all ( i 1 , . . . , i n ) satisfying i 1 ≤ · · · ≤ i L 1 > i L 1 + 1 ≤ · · · ≤ i L 1 + L 2 > · · · > i L 1 + ··· + L k − 1 + 1 ≤ · · · ≤ i n . � �� � � �� � � �� � L 1 L 2 L k • r L is the noncommutative generating function for words on the alphabet P with descent composition L . Example The words 221552 and 374443 have descent composition ( 2 , 3 , 1 ) , so X 2 2 X 1 X 2 5 X 2 and X 3 X 7 X 3 4 X 3 are both terms in r ( 2 , 3 , 1 ) . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Noncommutative Symmetric Functions • Let Sym n be the vector space with basis { r L } L � n . Then ∞ � Sym := Sym n n = 0 is a subalgebra of Q �� X 1 , X 2 , . . . �� called the algebra of noncommutative symmetric functions. Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Noncommutative Symmetric Functions • Let Sym n be the vector space with basis { r L } L � n . Then ∞ � Sym := Sym n n = 0 is a subalgebra of Q �� X 1 , X 2 , . . . �� called the algebra of noncommutative symmetric functions. • Noncommutative symmetric functions were formally introduced in 1995 by Gelfand, Krob, Lascoux, Leclerc, Retakh, and Thibon, but appeared implicitly in the 1977 Ph.D. thesis of Ira Gessel. Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Noncommutative Symmetric Functions • Let Sym n be the vector space with basis { r L } L � n . Then ∞ � Sym := Sym n n = 0 is a subalgebra of Q �� X 1 , X 2 , . . . �� called the algebra of noncommutative symmetric functions. • Noncommutative symmetric functions were formally introduced in 1995 by Gelfand, Krob, Lascoux, Leclerc, Retakh, and Thibon, but appeared implicitly in the 1977 Ph.D. thesis of Ira Gessel. � • Let h n := r ( n ) = X i 1 X i 2 · · · X i n . Then the h n are i 1 ≤ i 2 ≤···≤ i n algebraically independent and generate Sym . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Outline 1 Introduction 2 Three Basic Homomorphisms 3 Homomorphisms Arising from Shuffle-Compatibility Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Our Approach • Many permutation enumeration formulas involving runs or descents can be proven in the following way: 1 Derive a lifting of the formula in Sym . 2 Apply an appropriate homomorphism. Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility Our Approach • Many permutation enumeration formulas involving runs or descents can be proven in the following way: 1 Derive a lifting of the formula in Sym . 2 Apply an appropriate homomorphism. • The simplest such homomorphism is Φ: Sym → Q [[ x ]] defined by Φ( h n ) = x n n ! . Then Φ( r L ) = β ( L ) x n n ! where β ( L ) is the number of permutations with descent composition L � n . Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
Introduction Three Basic Homomorphisms Homomorphisms from Shuffle-Compatibility David and Barton’s Formula Lemma (Gessel & Z. 2014) � ∞ � − 1 � � r L = ( h mn − h mn + 1 ) n = 0 L all parts < m Homomorphisms on Noncommutative Symmetric Functions and Permutation Enumeration Yan Zhuang
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