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## The Algebra of Cycles

We have seen that the transport of structure along arrows for combinatorial species or classes does not transport structures but the underlying symmetry, and only the cyclic structure counts when counting.

The very same way the set operations are reducible to element-wise operations, the cycle index operations go cycle-wise.

The theory of Combinatorial Species is not a conspiration theory but it is deep rooted in Gear Algebra.

### Cyclewise operations (for cycle indices)

Addition means to consider more asynchronous cycle at a time.

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### Element-wise operations (with sets)

 Element-wise operations with sets

Addition means to consider more asynchronous cycle at a time.

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### 1-wise operations - Arithmetics

Sum and Difference go by asociativity

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### NOTES

2,3) If the cycles are synchronous, we get the multiplication and the cartesian product. These two operations work together like an usual ring.

3) The Cartesian Product is related to double labelling and equivalently, bipartite graphs. First seen at Harary (1958) it came into this page as the combinatorial variant of Boolean intersection.

3) Extensionality can not be removed from mathematics. Red balls and blue balls, North and East, apples and oranges are not removed but encoded in good old vectors under the names of *left*  and *right*.

Remember that (a, b) + (c, d) goes like a-with-c and b-with-d, i.e. *left with left* and *right with right*, exactly as we were summing two fruit baskets, apples with apples and oranges with oranges.

Here comes into place the cartesian product.

4) Amazing, the cyclewise composition is just

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{\displaystyle C\_{a}\circ C\_{\color {blue}b}=C\_{\color {blue}b.a}}

 How they did not see this one ? maybe they saw it, but it looks very commutative and, moreover, one has to explain what a cycle is 😂 The asymmetry occurs like this e.g.:

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{\displaystyle x\_{2}^{4}\circ (x\_{3}+x\_{4})=(x\_{2}\circ (x\_{3}+x\_{4}))^{4}=(x\_{6}+x\_{8})^{4}}

 Meanwhile,

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{\displaystyle (x\_{3}+x\_{4})\circ x\_{2}^{4}=x\_{3}\circ x\_{2}^{4}+x\_{4}\circ x\_{2}^{4}=x\_{6}^{4}+x\_{8}^{4}}

## Labelled Issues

```
When taking into account more sorts, there is a rule of thumb : For each exclamation mark that occurs in a combinatorial formula, there is one different sort that lies behind.

```

###  Sum and Product

* 2958701  ...9 different candies to three different kids
* 2839811  Suppose we throw 180 balls into 10 bins
* 3300469  Number of possible ways to pair elements from two datasets
* 2853761  Combinations involving distinct sets of variables
* 2855400  pick 3 balls out of 2 black and 3 white balls no-rep

### Constants: F(A,X) = A⋅F'(X)

* 2852683 how many subsets of size k from {1,2,…n} such that if subset contains 2 it doesn't contain 1
* 2990035 two parts of five, 1 and 2 in the same part.

### Hypergeometry

* 2845535 ...the sixth ball is white
* 3484751  ...ten gifts of three kinds
* 2845039  ...the last king appears on the 48th card?

### Counters

* 2814725 A group of 6 men and 6 women is randomly divided into 2 groups of size 6 each (Ross)
* 2821613 A closet contains 10 pairs of shoes (Ross)
* 2820643 18 days on which exams can take place, exams to take,
* 2822596 A basketball team consists of 6 frontcourt and 4 backcourt players.
* 2852248 Number of ways of forming 10 student committee from 5 classes of 30 students each

###  Ice Hockey -Mixed GF

* 3911810  combination-problem-picking-a-basketball-team-with-restrictions?
* 4533892  in-how-many-ways-can-the-ice-hockey-team-be-formed/4536045#4536045

### Combs

* 1493288  3 sorts. An urn contains six ball of each of the three colors: red, blue, and green.
* Poker Hands  ...13 sorts

### Composition

* 2810077  How many functions are there from [9] to [7] if every image in the codomain has 3 arguments in the domain?
* 104597  Six slots and six balls to allocate
* 2937270  36 groups of 5
* 2929506  Explicit Z, 4 apples, 5 oranges and 6 bananas in 4 baskets

## More Labelled Issues

### Differentiation

* 864116: decks when both rank and color matter Generalised Differentiation
* mse 1234919  richard-pavliceks-bridge Bruijn partial delabeling
* **Sampling** Let's say that John has 3 chickens and 2 ducks and he wants to remove 2 birds
* **Subtraction Tables** - Subtraction of groups

### Geometry

* **Transitivity** - Stick-Stuck, The Spectrum of Species, Carmichael
* **Coordinates** - Introducing coordinates, Labelling the Fano Plane
* **Pointing** - Fano', Chvatal', Prism', Cube'
* **Configurations in Fano Plane**

### Double Labelling - Two Dimensions

* **Boole** A sort X,Y may be like AS, ace of spade, an item that is both ace and spade. Boolean Intersection is a Cartesian Product.

Double Labelling or bipartite graphs.

```
https://mathworld.wolfram.com/BipartiteGraph.html
The numbers of connected bipartite graphs on , 2 ... nodes are 1, 1, 1, 3, 5, 17, 44, 182, ... (OEIS A005142)
https://www.sciencedirect.com/science/article/pii/S0012365X81800155

```

* mse 864116  How to count different card combinations with isomorphism? (2D)

### Functorial composition

* mse 498954  Counting number of distinct systems (2D)

```

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 {\displaystyle The\ Cylinder:\ \ \ X^{13}\times {\mathit {4}}(Y)={\mathit {4}}(Y^{13})}

```

* mse 1632337 How many 5-card hands are there with 3 hearts and a three-of-a-kind?

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 {\displaystyle {\begin{aligned}{\mathit {10}}(X)\cdot {\mathit {2}}(X)\cdot &[X,Y]\\&[X,Y]\cdot Y\cdot {\mathit {2}}(Y)\end{aligned}}}

```

## Unlabelled Issues

### Store and Inventory

* 3283154 A score of eleven in three throws
* 2976301 Spending euros each day

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{\displaystyle d^{k}.(e+e^{2}+e^{2})^{k}}

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{\displaystyle d^{k}e^{n}}
* 2820559 Distribute 10 things among 6 people (C. Blatter)
* 648649 What is the probability no slots contain more than two balls given I am trying to sort 5 balls into 6 slots?
* 2935168 Distributing 11 socks into 4 drawers, with 5 red socks and 6 blue socks.

###  Classic P.I.E.

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{\displaystyle AtLeast(t)=Exact(t+1)=}

 Each object is a multi-sort one

* 2916023

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{\displaystyle N(X,Y,Z) = 100 + 50[X] + 33[Y] + 20[Z] + 16[X,Y] + 10 [Y,Z] + 6[Y,Z] + 3[X,Y,Z]}

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* 2916023 choosing  cards out of -card deck so that there are exactly three different suits (Inclusion-Exclusion)]

###  Short Strings

* 3064844 - CALCULUS In how many ways can letters in a word CALCULUS be rearranged so that no two identical letters are adjacent?
* 2569399-RRRGGGBBB In how many ways can red, blue, and green balls be arranged
* 124452 - SLUMGULLION How many ways can the letters in the word SLUMGULLION
* 2698725 - AABBCCDDEE containing two of each where no two adjacent elements are alike
* 2955019 - RrGgBb  On the probability of placing cups on saucers

###  Soup and Croutons

* 2827898  Words of Length r over [n] with no three consecutive letters the same
* 295695  specified appearances of 101
* 2818572 specified appearances of 110
* 3297032  Probability of drawing a sequence of balls with at least consecutive red balls

### String Factories

* 2915770 How many binary strings of length n exist with k 10's or 01's
* 2940875 There are no words that contain sequences of s with length exactly 2 There are no words that contain sequences of s with length exactly 2 or 3.
* 2698725 The number of sequences of length that can be formed with unique digits containing two of each where no two adjacent elements are alike
* Goddard Binary strings with even number of 0's

## Notes

* Surjections
* Fibonacci as types of labeled structures
* **Second Degree Expresions**  Catalan Numbers, Parantheses, Seven Trees in One, Divergent Series
* **Formal Languages and Automata** - Counting strings in Peter Linz book
* **Polya proof**  Proof by the mean of colored permutations ; double permutations ; Bruijn
* **Coupons**  - Coupons without replacements
* **Sudoku** - There are only 9 labels in a Sudoku game
* Substructures extra Ditran-Tritran
* **Alexander's Paradoxes**

## Appendix

* Polya coloring : Cubes, Necklaces, Assymmetry  (Marko)
* Some Formulas
* Marks Tables
* Reviews

## Stuff

* Math Symbols
* Sandbox

* https://math.stackexchange.com/questions/1125878/a-starting-lineup-consists-of-2-forwards-2-guards-and-1-center-how-many-differ1
* https://math.stackexchange.com/questions/130126/basketball-team-combinatorial
* https://math.stackexchange.com/questions/4067964/in-how-many-ways-can-a-team-be-formed-when-some-players-can-play-multiple-positi

* https://math.stackexchange.com/questions/2924139/mixture-of-negative-binomial-distributions-technically-some-of-them-are-geometr
* https://math.stackexchange.com/questions/2938114/problem-about-eight-different-books-randomly-put-on-shelf
* https://math.stackexchange.com/questions/36022/how-do-the-catalan-numbers-turn-up-here?rq=1
* mse #3299910 what is the probability of drawing 2 blue pens and 1 black pen?

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{\displaystyle \left({\frac {4}{9}}b+{\frac {3}{9}}k+{\frac {2}{9}}r\right)^{3}}

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{\displaystyle b^{2}k^{1}}

* https://math.stackexchange.com/questions/4534085/spelling-bee-real-combinatorics-example