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Additional resources for An Introduction to Quasisymmetric Schur Functions (September 26, 2012)
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.
However, β α implies D(w, γ) = set(α) ⊆ set(β ). Consequently, we have Fα = ∑ xi1 · · · xin , where the sum is over all sequences (i1 , . . , in ) of positive integers satisfying i1 ··· in and i j < i j+1 if j ∈ D(w, γ). Now F(w, γ) = ∑ x f (w1 ) · · · x f (wn ) . 17, we have f ∈ O(w, γ) if and only if f (w1 ) ··· f (wn ) and f (w j ) < f (w j+1 ) if j ∈ D(w, γ). Since the values in the range of a (w, γ)-partition can be any positive integers, the result follows. 3 The Hopf algebra of quasisymmetric functions 39 We have just shown that every weight enumerator of a labelled chain is a fundamental quasisymmetric function.
3] or [31, Chapter 5], one of which we now give. 13 (Littlewood-Richardson rule). Let µ, ν be partitions. Then sν/µ = ∑ cνλ µ sλ where the sum is over all partitions λ , and the Littlewood-Richardson coefficient cνλ µ counts the number of SYTs (respectively SRTs) T of shape ν/µ such that us- 28 3 Hopf algebras ing Schensted (respectively reverse Schensted) insertion rect(wcol (T )) = Vλ (respectively Vˇλ ). 14. We have s(2,2,1)/(1) = s(2,2) + s(2,1,1) from the SYTs 3 1 4 4 1 2 3 2 with respective column reading words 3142 and 4132 whose respective rectifications are the following canonical SYTs.