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Extra resources for An Introduction to Quasisymmetric Schur Functions (September 26, 2012)
Now suppose that f is a (P, γ)-partition. Let w be the chain with underlying set P and order defined by: p < q in w if 1. f (p) < f (q) or 2. f (p) = f (q) and γ(p) < γ(q). 13 ensures that one of the two conditions is satisfied. Thus w is a linear extension of P, and it is clearly the only one for which f is a (w, γ)-partition. The converse is trivial: If w is a linear extension of P, then every (w, γ)-partition is also a (P, γ)-partition. 23. 24. If (P, γ) is a labelled poset, then its weight enumerator F(P, γ) = ∑ F(w, γ), where the sum is over all linear extensions w of P.
2. We shall use the following notation. Given a composition α = (α1 , . . , αk ) and a k-tuple I = (i1 , . . , ik ) of positive integers i1 < · · · < ik , let xIα α denote the monomial xiα11 · · · xikk . A quasisymmetric function is a formal power series f ∈ Q[[x1 , x2 , . ]] such that 1. f has finite degree, 32 3 Hopf algebras 2. f is invariant under the action of S∞ on Q[[x1 , x2 , . I is defined to be the k-tuple obtained by arranging the numbers σ (i1 ), . . , σ (ik ) in increasing order.
Then we define the composition corresponding to a path P with m steps, denoted by γP , to be γP = (γ1 , . . 11) where if the i-th step is (1, 0) and Pi−1 = (q − 1, r − 1) αq if the i-th step is (0, 1) and Pi−1 = (q − 1, r − 1) γi = βr αq + βr if the i-th step is (1, 1) and Pi−1 = (q − 1, r − 1). Equivalently, we define the composition γP = (γ1 , . . , γm ) corresponding to a path P with m steps recursively as follows. If the i-th step of P is (0, 1) or (1, 0), we let γi be the leftmost part of α or β , respectively, that has not been used previously to define a part of γP ; and if the i-th step is (1, 1), we let γi be the sum of the leftmost parts of α and β that have not been used previously.