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https://github.com/hivert/coq-combi

Algebraic Combinatorics in Coq
https://github.com/hivert/coq-combi

combinatorics coq coq-formalization mathcomp symmetric-functions

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Algebraic Combinatorics in Coq

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README 

        
Coq-Combi

=========



Formalisation of (algebraic) combinatorics in Coq/MathComp.



Authors

========================================================================



Florent Hivert 



Contributors:



- Thibaut Benjamin (representation theory of the symmetric groups)

- Jean Christophe Filliâtre (Why3 implementation)

- Christine Paulin (SSreflect binding for ALEA + hook length formula)

- Olivier Stietel (hook length formula)



This library was supported by additional discussions with:



- Georges Gonthier

- Assia Mahoubi

- Pierre Yves Strub

- Cyril Cohen

- the SSReflect mailing list



The project was transferred to mathcomp on 2021-10-20.



Contents

========================================================================



* basic **theory of symmetric functions** including



  - *Schur function* and *Kostka numbers* and the equivalence of the

    combinatorial and algebraic (Jacobi) definition of Schur polynomials



  - the classical bases, *Newton formulas* and various basis changes



  - the scalar product and the *Cauchy formula*



* the **Littlewood-Richardson** rule using Schützeberger approach, it includes



  - the *Robinson-Schensted* correspondance



  - the construction of the *plactic monoïd*



  - the *Littlewood-Richardson* and *Pieri* rules using the combinatorial

    (tableau) definition of Schur polynomials.



  After A. Lascoux, B. Leclerc and J.-Y. Thibon, "The Plactic Monoid" in

  Lothaire, M. (2011), Algebraic combinatorics on words, Cambridge University

  Press With variant described in G. Duchamp, F. Hivert, and J.-Y. Thibon,

  Noncommutative symmetric functions VI. Free quasi-symmetric functions and

  related algebras. Internat. J. Algebra Comput. 12 (2002), 671–717.



* the **Murnaghan-Nakayama** rule for converting power sum to Schur function,

  it includes



  - two recursive implementations building the tableau up or down



  - a skew version multiplying a Schur function by a power sum expanding the

    result on Schur functions.



* the **character theory of the symmetric Groups**. We do not use

  representations but rather goes as fast as possible to Frobenius

  isomorphism and then uses computations with symmetric polynomials. it includes



  - *cycle types* for permutations (together with Thibaut Benjamin)



  - The tower structure and the *restriction and induction formulas* for class

    indicator (together with Thibaut Benjamin)



  - structure of the *centralizer* of a permutation



  - Young character and *Young Rule*



  - the theory of Frobenius characteristic and *Frobenius character formula*



  - the *Murnaghan-Nakayama* rule for evaluating irreducible characters



  - the *Littlewood-Richardson* rule for irreducible characters



* the **Hook-Length Formula** for standard Young tableaux

  (together with Christine Paulin and Olivier Stietel). We follow closely



   Greene, C., Nijenhuis, A. and Wilf, H. S. (1979) A probabilistic proof of a

   formula for the number of Young tableaux of a given shape. Adv. in

   Math. 31, 104–109.



* the **Erdös Szekeres theorem** about increassing and decreassing subsequences



   from Greene's invariants theorem.



* various **Combinatorial objects** including



  - integer partitions and compositions,

  - skew partition, horizontal, vertical and ribbon border strip,

  - tableaux, standard tableaux, skew tableaux,

  - subsequences, integer vectors,

  - standard words and permutations,

  - Yamanouchi words,

  - binary trees, Dyck words



* the **Coxeter presentation of the symmetric group**.



  We formalize:



  - presentation of the symmetric group generated by elementary

    transpositions

  - Matsumoto theorem saying that two reduced words give the same permutation

    iff they are equivalent under braid relations

  - the Coxeter length and the inversion set

  - the dual Lehmer code of a permutation

  - the weak permutohedron lattice



* the **factorization** of the Vandermonde determinant as the product

  of differences.



* the **Tamari lattice** on binary trees.



* the formula for **Catalan numbers** counting binary trees and Dyck words.



  I use a bijective proof using rotations. There is a generating function

  proof available in https://github.com/hivert/FormalPowerSeries which I plan

  to merge here at some points.



Various unstable/unfinished experiments:

========================================



* bijection m-trees <-> m-dyck words.

  See the [trees branch on Github](https://github.com/hivert/Coq-Combi/tree/trees).



* a **Why3 certified implementation** of the LR-Rule

  (together with Jean Christophe Filliâtre).

  See the [Why3 branch on Github](https://github.com/hivert/Coq-Combi/tree/Why3).



* Poset.

  See the [posets branch on Github](https://github.com/hivert/Coq-Combi/tree/posets).



* Formal Power series. See the series branch on Github.

  See the [series branch on Github](https://github.com/hivert/Coq-Combi/tree/series).



* Set-partitions

  See the [SetPartition branch on Github](https://github.com/hivert/Coq-Combi/tree/SetPartition).



Documentation

========================================================================



* The [documentation](http://hivert.github.io/Coq-Combi/) is now complete !



* A

  [presentation](https://github.com/hivert/Coq-Combi/raw/master/doc/Talk-CRM/CRM.pdf)

  given at "[Algebra and combinatorics at LaCIM](http://www.crm.umontreal.ca/2018/Algebre18/index_e.php), a conference

  for the 50th anniversary of the CRM", September 24-28, 2018, Montreal,

  Quebec, Canada. This presentation is targeted at combinatorialist.



* Another

  [presentation](https://github.com/hivert/Coq-Combi/raw/master/doc/Talk/INRIA.pdf)

  given at [Specfun](https://specfun.inria.fr/seminar/) Inria seminar, march

  2015. This presentation is targeted at proof-assistant specialist.



Installation

========================================================================



This library is based on



* Coq 8.13 or more recent



* [SSReflect/MathComp]([https://github.com/math-comp/math-comp])

  Library version 1.12 or more recent.



* Pierre-Yves Strub library for

  [Multinomials] version 1.5.4 (https://github.com/math-comp/multinomials)



Here are the Opam packages I'm using

```

coq-mathcomp-ssreflect    1.12.0

coq-mathcomp-algebra      1.12.0

coq-mathcomp-field        1.12.0

coq-mathcomp-fingroup     1.12.0

coq-mathcomp-character    1.12.0

coq-mathcomp-multinomials 1.5.4

```