For this subject matter, definitions and even formal calculations and heuristic deductions seem to me often more important than complete proofs. Polya, 1937
Why e.g.f.'s ? Edit
A G.F. is always produced by an e.g.f. When aggregating two solids, the exponential stuff vanishes like in
$ 3! {x^3 \over 3!} . 7! {x^7 \over 7!} = 10! {x^{10} \over 10!} $ that becomes $ x^3 . x^7 = x^{10} $
The aggregate operation rule is $ 3! * 7! = (3+7)! $ and it is discarded.
Current work Edit
- Goulden-Jackson: N(x)=E(x+1)
- Grammars - Unambiguous grammars are combinatorial species.
- Exclusion: $ F(X+Y-XYX + XYXYX) $
Welcome to Species : the ArticleEdit
Combinatorial species at work - the article, original research !
OperationsEdit
- Summation: $ F(X) + G(X) = F(X) + G(Y) $
- Constants and the double reading of F(A,X)
- Composition Wreath product, Plethistic substitution
- Differentiation : $ F' $ and point stabilizers: $ Chvatal', \ Prism', \ Cube' $, multisort differentiation, stick-k-stuck-n
- Sampling: generalized differentiation
- Multisorting: F(X+Y)
- Merging sorts - how un-coloring does or not affect cardinality
- Weighted species = the t counter
- Functorial Composition - card decks, shoes, $ \bbox[border:1px solid black]{^AS} $
- Hypergeometric: The sixth ball is white
Alexander's paradoxes Train, cafeteria
Divergent series or linear species.
More examples Sudoku
Appendix Edit
Subtraction : 6-2, 4-2
Fano plane : subspecies of Fano Plane
https://oeis.org/A011971 to see
Solomon, L. (1967), "The Burnside algebra of a finite group", J. Comb. Theory, 1: 603–615
StuffEdit
Let $A_n$ be the number of such sequences ending in `A`, and $B_n$ be the number of such sequences ending in `B`. We count the empty string in both.
$ \bbox[1px,border:2px solid #00A000]{^AS} $
https://math.stackexchange.com/questions/2828924/coloring-cube-sides-and-vertices
$ \bbox[5px,border:2px solid #00A000]{ \begin{gather} H_X(N, M) = \frac{N! \times M!}{24} \\ \times \left({6\brace N} {8\brace M} + 6 {3\brace N} {2\brace M} + 3 {4\brace N} {4\brace M} + 8 {2\brace N} {4\brace M} + 6 {3\brace N} {4\brace M}\right). \end{gather}} $
$ \sum_{S \subset N \setminus \{i\}} {|S|!(n-|S|-1)!}=n! $
https://math.stackexchange.com/questions/2855911/combinatorial-proof-of-negative-binomial-identity
$ \begin{eqnarray*} \underbrace{** }_{ k_1 \text{ stars}} \mid \underbrace{*** }_{ k_2 \text{ stars}} \mid \cdots \mid \underbrace{*}_{ k_n \text{ stars}} \\ \sum_{i=1}^{n} k_i=k. \end{eqnarray*} $
ExtraEdit
Dytran-threetran will it ever find its place ?
https://math.stackexchange.com/questions/2912210/another-beauty-hidden-in-a-simple-triangle-3
Latest ActivityEdit
- Grammars
edited by BoykuEdited the section: SLUMGULLION - Weighted species
edited by BoykuEdited the section: The game of Clue involves 6 suspects, 6 weapons, and 9 rooms (Ross) - Functorial Composition
edited by BoykuSummary: an urn contains six ball of each of the three colors: red, blue, and green: - Functorial Composition
edited by BoykuEdited the section: Two pair is 123552 - Home
edited by BoykuEdited the section: Operations - Hands and Decks
edited by BoykuEdited the section: Two pair is 123552 - Hands and Decks
edited by BoykuEdited the section: Poker hands - Hands and Decks
edited by BoykuEdited the section: Choose one - 52 - Hands and Decks
edited by BoykuEdited the section: Two pair is 123552 - Goulden-Jackson
edited by BoykuEdited the section: The number of sequences of length 10 that can be formed with 5 unique digits containing two of each where no two adjacent elements are alike
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