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Epistemology - m a t h 4 w i s d o m - g m a i l
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Introduction E9F5FC Questions FFFFC0 Software |
- How does the expansion (x1 + ... + xm)N relate to the matrix of nonnegative integers? and how it yields pairs of Kostka matrices? (form and content)?
The factoring (number of simplexes n choose k - dependent simplex) x (number of flags on k - independent Euclidean) x (number of flags on n-k - independent Euclidean) = (number of flags on n) The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). Conjugation gives the ways of relabeling, renaming. For example, (132)(12)(123) relables 1 as 2 and 2 as 3 in (12) to get (23). I want to list and generate the basic combinatorial objects. Stanley Enumerative Combinatorics - Volume I
- Volume II: Table of Contents
- The Twelvefold Way
- Permutations
- Sieve methods - Inclusion Exclusion
- Partially ordered sets
- Rational generating functions
- Trees and the Composition of Generating Functions
- Algebraic, D-Finite, and Noncommutative Generating Functions
- Symmetric functions
The Twelvefold Way f:N->X two sets - f is arbitrary (no restriction)
- f is injective (one-to-one)
- f is surjective (onto)
And regarding the elements of N and X as "distinguishable" or "indistinguishable". {$(x)_{n}=x(x-1)(x-2)...(x-n+1)$} {$S(n,k)$} is the number of partitions of an n-set into k-blocks. It is called a Stirling number of the second kind. |

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