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**Sample text**

4 The Hopf algebra of noncommutative symmetric functions Our third and final Hopf algebra involves noncommutative symmetric functions, defined by Gelfand et al. in [34]. As we will see, they are closely connected to both quasisymmetric and symmetric functions. Throughout we use the notation used in [11], which evokes the relationship with symmetric functions. 1. The Hopf algebra of noncommutative symmetric functions, denoted by NSym, is Q e1 , e2 , . . generated by noncommuting indeterminates en of degree n with the operations of the next subsection.

Given two permutations σ = σ (1) · · · σ (n) ∈ Sn and τ = τ(1) · · · τ(m) ∈ Sm , we say a shuffle of σ and τ is a permutation in Sn+m such that σ (i) appears to the right of σ (i − 1) and to the left of σ (i + 1) for all 2 i n − 1 and similarly, τ(i) + n appears to the right of τ(i − 1) + n and to the left of τ(i + 1) + n for all 2 i m − 1. We denote by σ τ the set of all shuffles of σ and τ. 8. 12 Let α Then n, β ✁ 21 = {1243, 1423, 1432, 4123, 4132, 4312}. m and σ ∈ Sn , τ ∈ Sm , such that d(σ ) = set(α) and d(τ) = set(β ).

Let λ = (5, 4, 2, 2). Below is the Young diagram of λ , with the label γλ (i, j) written in the cell with coordinate pair (i, j). 40 3 Hopf algebras 1 5 2 6 3 7 9 4 8 10 12 11 13 Let T be an SSYT of shape λ . If we regard T as the map from Pλ to the positive integers that sends the pair (i, j) to the entry of the cell with coordinate pair (i, j), then T is a (Pλ , γλ )-partition. We now resume our development of the theory of P-partitions. The next lemma will allow us to write any weight enumerator as a sum of weight enumerators of chains.