The symmetric functions catalog

The term equation is not really accurate, as these are really identities. However, the general public seem to use the term equation to mean a bunch of math symbols with an equality sign somewhere.

I take no responsibility for regrets if you decide to get a tattoo of any of these — but if you get one, send me a picture!

#\(q\)-Catalan refinement

We have the identity \[\catalan(n-2;q) = \sum_{k \geq 0} q^{k(k-2)} \frac{[n]_q}{[k]_q} \qbinom{n-4}{2k-4}_q\catalan(k-2;q) \left( \sum_{j=0}^{n-2k} q^{j(n-2)}\qbinom{n-2k}{j}_q\right).\]

This is a refinement of the \(q\)-Catalan numbers.

#Unicellular LLT \(e\)-expansion

I conjectured, and a few years later together with R. Sulzgruber [AS20], proved the following formula for unicellular LLT polynomials in the elementary symmetric functions basis.

For any unit-interval graph \(\avec\) with \(n\) vertices we have the expansion \[\LLT_\avec(\xvec;q+1) = \sum_{\theta \in O(\avec)} q^{\asc(\theta)}\elementaryE_{\pi(\theta)}(\xvec).\]

#Border-strip decompositions of a rectangle

Let \(a(n)\) be the number of ways to tile a \(2n \times n\) rectangle with border-strips of size \(n.\) Then with \(a(0)=1,\) we have \[a(n) = \frac{1}{2} \sum_{i=1}^n \frac{ i(n - i + 1)}{ (n + 2)} \binom{n - 1}{i - 1} \binom{n + 3}{i + 1} a(i - 1) a(n - i).\] This is proved in [AJ19]. See also A115047.

#Circular Dyck diagrams

In [ALP19], we find the major-index generating function for circular Dyck diagrams of size \(n\) and width at most \(w.\) \[|\CDP(n,w)|_q = \sum_{s \in \setZ} \sum_{j=1}^w q^{s^2\delta + s(j+1)} \left( \qbinom{2n-1}{n-1-\delta s}_q - \qbinom{2n-1}{n+j+\delta s}_q \right),\] where \(\delta = w+2.\)

See also A194460.

#Normalized Jack–Kostka coefficients

This is from [Thm. 5.12, AF17]. For \(\lambda \vdash n,\) let \[\mathfrak{K}_{(k)}^{(\alpha)}(\lambda) \coloneqq \frac{1}{(n-k)!} [\monomial_{k,1^{n-k}}] \jackJ_\lambda(\xvec;\alpha).\] which is a normalized hook coefficient in the monomial basis of the Jack J symmetric function.

Then \[\mathfrak{K}_{(k)}^{(\alpha)}(\lambda) = \sum_{\substack{ A \subseteq \lambda, \ |A|=k \\ \text{column-distinct}}} \prod_{\substack{R \text{ row} \\ \text{of }\lambda}} P_{|R \cap A|}(\alpha) ,\] where \(P_i(\alpha) \coloneqq \prod_{j=0}^{i-1} (1 + j \, \alpha).\)

This equation allows us to see that \(\mathfrak{K}_{(k)}^{(\alpha)}(\mathbf{r}^{\mathbf{p}})\) is non-negative in the falling factorial basis (where we use multi-rectangular coordinates).

Bibliography

1. [AF17]Per Alexandersson and Valentin Féray. Shifted symmetric functions and multirectangular coordinates of Young diagrams. Journal of Algebra, 483:262–305, August 2017.
   .bib

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   @article{AlexanderssonFeray2017,
     doi = {10.1016/j.jalgebra.2017.03.036},
     url2 = {https://doi.org/10.1016/j.jalgebra.2017.03.036},
     year  = {2017},
     month = aug,
     publisher = {Elsevier {BV}},
     volume = {483},
     pages = {262--305},
     author = {Per Alexandersson and Valentin F{\'{e}}ray},
     title = {Shifted symmetric functions and multirectangular coordinates of {Y}oung diagrams},
     journal = {Journal of Algebra}
   }
   

2. [AJ19]Per Alexandersson and Linus Jordan. Enumeration of border-strip decompositions. Journal of Integer Sequences, 22(4):1–20, 2019.
   .bib

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   @article{AlexanderssonJordan2018,
   Author = {Per Alexandersson and Linus Jordan},
   Title = {Enumeration of border-strip decompositions},
   Year = {2019},
   journal = {Journal of Integer Sequences},
   volume = {22},
   number = {4},
   pages = {1--20},
   paper = {5},
   issn = {1530-7638},
   url = {https://cs.uwaterloo.ca/journals/JIS/VOL22/Alexandersson/alex4.html}
   }
   

3. [ALP19]Per Alexandersson, Svante Linusson and Samu Potka. The cyclic sieving phenomenon on circular Dyck paths. The Electronic Journal of Combinatorics, 26:1–32, October 2019.
   .bib

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   @article{AlexanderssonLinussonPotka2019,
     doi = {10.37236/8720},
     url2 = {https://doi.org/10.37236/8720},
     year = {2019},
     month = oct,
     publisher = {The Electronic Journal of Combinatorics},
     author = {Per Alexandersson and Svante Linusson and Samu Potka},
     title = {The Cyclic Sieving Phenomenon on Circular {D}yck Paths},
     journal = {The Electronic Journal of Combinatorics},
     volume = {26},
     paper = {P4.16},
     pages = {1--32}
   }
   

4. [AS20]Per Alexandersson and Robin Sulzgruber. A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions. arXiv:2004.09198, 2020.
   .bib

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   @article{AlexanderssonSulzgruber2020x,
   Author = {Per Alexandersson and Robin Sulzgruber},
   Title = {A combinatorial expansion of vertical-strip {LLT} polynomials in the basis of elementary symmetric functions},
   Year = {2020},
   Eprint = {2004.09198},
   url = {https://arxiv.org/abs/2004.09198},
   journal = {arXiv e-prints}
   }