By Jeffrey Remmel, Visit Amazon's Anthony Mendes Page, search results, Learn about Author Central, Anthony Mendes,
This monograph offers a self-contained creation to symmetric services and their use in enumerative combinatorics. it's the first e-book to discover some of the equipment and effects that the authors current. various routines are integrated all through, besides complete recommendations, to demonstrate suggestions and in addition spotlight many attention-grabbing mathematical ideas.
The textual content starts off through introducing basic combinatorial gadgets resembling variations and integer walls, in addition to producing functions. Symmetric services are thought of within the subsequent bankruptcy, with a different emphasis at the combinatorics of the transition matrices among bases of symmetric functions. bankruptcy three makes use of this introductory fabric to explain how to define an collection of producing features for permutation data, after which those ideas are prolonged to discover producing features for numerous gadgets in bankruptcy 4. the subsequent chapters current the Robinson-Schensted-Knuth set of rules and a mode for proving Pólya’s enumeration theorem utilizing symmetric functions. Chapters 7 and eight are extra really expert than the previous ones, protecting consecutive development fits in diversifications, phrases, cycles, and alternating variations and introducing the reciprocity approach with a purpose to outline ring homomorphisms with fascinating properties.
Counting with Symmetric Functions will attract graduate scholars and researchers in arithmetic or comparable matters who're drawn to counting equipment, producing capabilities, or symmetric functions. the original strategy taken and effects and workouts explored by means of the authors make it a tremendous contribution to the mathematical literature.
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Extra resources for Counting with Symmetric Functions
Given λ n, we will count the number of ways can we form the monomial λ x1λ1 · · · xk k by multiplying out eµ = eµ1 · · · eµ by organizing our work into a table where rows are indexed by x1 , . . , xk and columns are indexed by eµ1 , . . , eµ . Place a 1 in the xi row and eµ j column entry of the table if the monomial selected from eµ j to contribute to a final product contains xi and place a 0 in the table otherwise. For example, when λ = (32 , 2, 12 ) and µ = (32 , 22 ), one possible table is x1 x2 x3 x4 x5 e3 e3 e2 e2 1 1 0 0 1 1 1 1 0 0 0 1 0 1 0 1 0 1 .
Proof. The left hand side of this equation is the signed, weighted sum over all pairs of the form (S, T ) where S is a column strict tableau of shape 1i , T is a column strict tableau of shape (1k , n − i) with n − i ≥ 1, the sign is (−1)i , and the weight is w(S)w(T ). 4: if the topmost integer in S is not smaller than the rightmost in T , move this integer from S to T . Otherwise, undo this operation. Since we require that n − i ≥ 1, there are fixed points which cannot be changed by this involution.
4. 4. The generating functions E(z) and H(z) for the elementary and homogeneous symmetric polynomials satisfy H(z) = 1/E(−z). ∞ 1 1 = 1/E(−z). = ∞ ∏i=1 (1 + xi (−z)) i=1 1 − xi z Proof. 4 as 1 = H(z)E(−z), we find 1= ∞ n n=0 i=0 ∑ ∑ (−1)i ei hn−i zn . Comparing coefficients of zn shows that ∑ni=0 (−1)i ei hn−i is equal to 0 for all n ≥ 1. In following the philosophy of providing simple combinatorial proofs whenever reasonable, we will prove this fact with a sign reversing involution on pairs of column strict tableaux.