The symmetric functions catalog

#More resources

For an introduction to the topic of symmetric functions and algebraic combinatorics, I recommend the following books. R. Stanley, Enumerative Combinatorics II [Sta01], L. Butler, Subgroup Lattices and Symmetric Functions [But94], B. Sagan, The Symmetric Group [Sag01], W. Fulton, Young Tableaux [Ful97] and I. Macdonald, Symmetric functions and Hall polynomials [Mac95].

Moreover, D. Bressoud — Proofs and confirmations: the story of the alternating sign matrix conjecture [Bre99], is a very nice reading. A. M. Foley, A. H. Morales, A. Rattan, and K. Yeats survey the work of I. P. Goulden and D. M. Jackson, including their contributions to classical enumeration, symmetric functions, permutation factorizations, and algebraic structures in quantum field theory [FMRY22]. I. Pak gives a broad survey of what it means for a formula to have a combinatorial interpretation, with many examples, open problems, and a complexity-theoretic perspective [Pak24].

#Recommended

• R. Stanley — How the upper bound conjecture was proved

• J. Haglund — The genesis of the Macdonald polynomial statistic

• R. Stanley — Enumerative and Algebraic combinatorics in the 1960’s and 1970’s.

• P. Woit — Topics in Representation Theory: The Weyl Integral and Character Formulas

• X. Viennot — Algebraic and bijective combinatorics course videos.

#Useful links

• Combinatorics Online — A discord server for people in combinatorics.

• Mathematical gemstones by M. Gillespie.

• Overview of generalized Kostka Polynomials.

• Grammar and math by D.B. West.

• A primer of mathematical writing by S. G. Krantz.

• arXiv Companion tools, including DOI and arXiv BibTeX helpers.

• Overleaf Learn, with short LaTeX references and examples.

#Software resources

• OEIS — database of integer sequences.

• Findstat — database and search engine for combinatorial statistics and maps.

• permutation pattern database by B. Tenner.

• Mathematica packages — Packages for Catalan objects, symmetric functions, permutation statistics, tableaux, \(q\)-analogs, and related computations.

• polynomial-tools — Exact univariate polynomial tools for real-rootedness, interlacing, and combinatorial experiments.

• RealRooted — An experimental Lean 4 library for real-rooted polynomials, interlacing, compatibility, and Pólya-frequency arguments.

• kostka — Rust code for Kostka coefficients, Ehrhart polynomials of Gelfand–Tsetlin polytopes, and \(h^*\)-vectors.

• lrcalc-rs — A Rust implementation with a liblrcalc-compatible interface for Littlewood–Richardson coefficient computations.

• arxiv.symmetricfunctions.com — Source code for the arXiv companion, including combinatorics browsing, DOI discovery, and BibTeX helper tools.

Bibliography

1. [Bre99]David Bressoud. Proofs and confirmations: The story of the alternating sign matrix conjecture. Cambridge University Press, 1999.
   .bib

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   @Book{Bressoud1999,
    author = {David Bressoud},
    title = {Proofs and confirmations: the story of the alternating sign matrix conjecture},
    publisher = {Cambridge University Press},
    year = {1999},
    doi = {10.1017/CBO9780511613449},
    address = {Cambridge New York},
    isbn = {9780511613449}
    }
   

2. [But94]Lynne M. Butler. Subgroup lattices and symmetric functions. American Mathematical Society, 1994.
   .bib

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   @book{Butler1994,
     Author = {Lynne M. Butler},
     Title = {Subgroup Lattices and Symmetric Functions},
     Publisher = {American Mathematical Society},
     url = {https://bookstore.ams.org/memo-112-539},
     Year = {1994},
     ISBN = {082182600X}
   }
   

3. [FMRY22]Angèle M. Foley, Alejandro H. Morales, Amarpreet Rattan and Karen Yeats. Combinatorial and Algebraic Enumeration: A survey of the work of Ian P. Goulden and David M. Jackson. Algebraic Combinatorics, 5(6):1205–1226, 2022.
   .bib

   Copy Download

   @article{FoleyMoralesRattanYeats2022,
     author = {Foley, Angèle M. and Morales, Alejandro H. and Rattan, Amarpreet
       and Yeats, Karen},
     title = {Combinatorial and {A}lgebraic {E}numeration: a survey of the work
       of {I}an {P}. {G}oulden and {D}avid {M}. {J}ackson},
     year = {2022},
     journal = {Algebraic Combinatorics},
     volume = {5},
     number = {6},
     pages = {1205--1226},
     publisher = {MathDoc/Centre Mersenne},
     doi = {10.5802/alco.269},
     url = {https://doi.org/10.5802/alco.269},
     eprint = {2209.10075}
   }
   

4. [Ful97]William Fulton. Young tableaux: With applications to representation theory and geometry. London mathematical society student texts (book 35). Cambridge University Press, 1997.
   .bib

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   @book{Fulton1997,
     title = {Young Tableaux: With Applications to Representation Theory and Geometry},
     doi = {10.1017/cbo9780511626241},
     author = {William Fulton},
     isbn = {978-0511626241},
     series = {London Mathematical Society Student Texts (Book 35)},
     year = {1997},
     publisher = {Cambridge University Press}
   }
   

5. [Mac95]Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
   .bib

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   @book{Macdonald1995,
   	address = {New York},
   	author = {Ian G. Macdonald},
   	edition = {Second},
   	isbn = {0-19-853489-2},
   	mrclass = {05E05 (05-02 20C30 20C33 20K01 33C80 33D80)},
   	note = {With contributions by A. Zelevinsky, Oxford Science Publications},
   	pages = {x+475},
   	publisher = {The Clarendon Press, Oxford University Press},
   	series = {Oxford Mathematical Monographs},
   	title = {Symmetric functions and {H}all polynomials},
   	year = {1995}
   }
   

6. [Pak24]Igor Pak. What is a combinatorial interpretation?. Open problems in algebraic combinatorics, 110:191–260, 2024.
   .bib

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   @incollection{Pak2024,
     author = {Igor Pak},
     title = {What is a combinatorial interpretation?},
     year = {2024},
     booktitle = {Open Problems in Algebraic Combinatorics},
     series = {Proceedings of Symposia in Pure Mathematics},
     volume = {110},
     pages = {191--260},
     publisher = {American Mathematical Society},
     doi = {10.1090/pspum/110/02007},
     url = {https://doi.org/10.1090/pspum/110/02007},
     eprint = {2209.06142}
   }
   

7. [Sag01]Bruce E. Sagan. The symmetric group. Springer New York, 2001.
   .bib

   Copy Download

   @book{Sagan2001,
     doi = {10.1007/978-1-4757-6804-6},
     url2 = {https://doi.org/10.1007/978-1-4757-6804-6},
     year = {2001},
     publisher = {Springer New York},
     author = {Bruce E. Sagan},
     title = {The Symmetric Group}
   }
   

8. [Sta01]Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First, 2001.
   .bib

   Copy Download

   @book{StanleyEC2,
       author = {Richard P. Stanley},
       title = {Enumerative {C}ombinatorics: {V}olume 2},
       publisher = {Cambridge University Press},
       year = {2001},
       edition = {First},
       isbn = {0521789877},
       doi = {10.1017/CBO9780511609589}
   }