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From "Noncrossing partitions in surprising locations" by Jon McCammond:

Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the modular group. In this article, the focus is on a lesser known example: the noncrossing partition lattice. The focus of the article is a gentle introduction to the lattice itself in three of its many guises: as a way to encode parking functions, as a key part of the foundations of noncommutative probability, and as a building block for a contractible space acted on by a braid group.

The noncrossing partition lattice is a relative newcomer to the mathematical world. First defined and studied by Germain Kreweras in 1972 , it caught the imagination of combinatorialists beginning in the 1980s and has come to be regarded as one of the standard objects in the field. In recent years it has also played a role in areas as diverse as low-dimensional topology and geometric group theory as well as the noncommutative version of probability. Due no doubt to its recent vintage, it is less well-known to the mathematical community at large than perhaps it deserves to be, but hopefully this short paper will help to remedy this state of affairs."

Through the years, I've noted some guises of the NCPs:

  1. Refined partitions of a polygon by nonintersecting line segments drawn between its vertices. These are enumerated in OEIS A134264, a natural refinement of the Narayana numbers A001263, whose row sums are the Catalan numbers A000108. (Similar to transforming $x$ to $x+1$ in the Narayana polynomials to obtain the f-polynomials for the associahedra, if $h_0$ is replaced by $x/(1+x)$ and the other $h_n$ by $( 1+x)^n$, the reversed f-polynomials (A033282, A086810) for the associahedra are obtained from the partition polynomials of A134264.)

  2. Appropriately marked NCPs generate the partition polynomials that give the coefficients of the formal power series $f^{(-1)}(x)$ that is the compositional inverse of a formal power series $f(x)$, in terms of the coefficients of the power series for the shifted reciprocal $x/f(x)$ .

  3. Number of Dyck paths of semilength $n$ whose ascent lengths form the $k$-th partition of the integer $n$ (cf. A125181).

  4. Number of ordered trees with $n$ edges whose node degrees form the $k$-th partition of the integer n  (cf A125181)

  5. The $n$-th NCP partition polynomial multiplied by $n$ gives the number of terms in the homogeneous symmetric monomials generated by $[x(1) + x(2) + ... + x(n+1)]^n$ under the umbral mapping $x(m)^j = h_j ,$ for any $m$. This connects compositional inversion, Sheffer Appell sequences, and the Hirzebruch criterion for the Todd class.

  6. The sequence of NCP polynomials form an Appell sequence as polynomials in a distinguished indeterminate, a special type of Sheffer polynomial sequence in the same family as the Bernoulli polynomials.

  7. The NCP polynomials give the moments associated to the free cumulants in free probability theory. (Cf. "Three lectures on free probability" by Novak and LaCroix, and the Ebrahimi-Fard and Patras refs in A134264.)

  8. There is a bijection between NCPs and primitive parking functions. (See Lemma 3.8 on p. 28 of Rattan, "Parking functions and other combinatorial structures.)

  9. In the combinatorial Hopf algebras of symmetric polynomials/functions, Sym, presented in "Hopf algebras and the logarithm of the S-transform in free probability" by Mastnak and Nica, the polynomials $y_n$ are the NCPs with the elementary symmetric polynomials (ESPs, denoted by $e_n$) as their indeterminates. The relationship between the o.g.f., $E(x)$, of the ESPs and the o.g.f., $H(x)$, of the complete homogeneous symmetric polynomials (CSPs, denoted by $h_n$) is $\frac{x}{xH(-x)}=E(x),$ and, since the $y_n$ by A134264 (link above) are also the coefficients of the inverse of $xH(-x)$, the $y_n$ when expressed in terms of the CSPs (cf. A263633) become the normalized signed partition polynomials of A133437 (with an index shift in the indeterminates and overall sign change for every other polynomial). These are the refined, signed face polynomials of the associahedra, the antipode of a Faa-di-Bruno-like Hopf algebra based on the refined Lah polynomials (A130561, normalized elementary Schur polynomials) and inversion of o.g.f.s rather than the refined Bell polynomials (A036040) and inversion of e.g.f.s. (More on these types of Hopf algebras in the MO-Q "Guises of the Stasheff polytopes" and "Hopf monoids and generalized permutahedra" by Aguiar and Ardila.)

  10. (Added May 2022) Relations to scattering amplitudes in quantum field theories are presented in "Connecting Scalar Amplitudes using The Positive Tropical Grassmannian" by Cachazo and Umbert, extending earlier work by Brezin, Itzykson, Parisi, and Zuber in "Planar Diagrams" (see A338135).

  11. (Added Feb 21, 2023) The set of NCP polynomials $$[N] = [1,N_1(u_1),N_2(u_1,u_2),...] = [1,u_1,u_1^2+u_2, u_1^3+3u_1u_2+u_3,...]$$ is the raising operator under substitution for the $m$-associahedra partition polynomials $$[A^{(m}] = [1,A^{(m)}_1(u_1),A^{(m)}_2(u_1,u_2),...],$$ which are the nonzero coefficients of the power series coefficients of the compositional inverse / series reversion of the formal o.g.f.s $$O^{(m)}(x) = x + c_1x^{m+1} + c_2 x^{2m+1} + c_3 x^{3m+1} + \cdots.$$ That is, $$N_n(A^{(m-1)}_1(u_1),A^{(m-1)}_2(u_1,u_2),...,A^{(m-1)}_n(u_1,...u_n)) =A^{(m)}_n(u_1,u_2,...,u_n),$$ or $$[A^{(m)}] = [N][A^{(m-1)}] = [N]^m[R]$$ with the starting point $$[A^{(1)}] = [A] = [N][R]$$ where $$[A] = [1,A_1(u_1),A_2(u_1,u_2),...] = [1,-u_1,2u_1^2-u_2,-5u_1^3 + 5u_1u_2 -u_3, ...]$$ are the associahedra partition polynomials of normalized, re-indexed A133437 for compositional inversion of o.g.f.s and $$[R] = [1,R_1(u_1),R_2(u_1,u_2),...] = [1,-u_1, u_1^2 - u_2,-u_1^3 + 2 u_1 u_2 - u_3,...]$$ are the reciprocal partition polynomials (refined Pascal polynomials) of signed A263633 for multiplicative inversion defined by the shifted reciprocal $$x/O(x)= \frac{1}{1+c_1x + c_2 x^2 + \cdots} = 1 + R_1(c_1) x + R_2(c_1,c_2) c^2 + \cdots.$$ This formalism for $[N^{(m)}] = [N]^m$, the refined $m$-Narayana / $m$-Noncrossing partition polynomials, is ensconced between that for the reduced versions of these polynomials, the mono-variable $m$-Narayana polynomials, as portrayed by Novelli and Thibon in "Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions" and that for the even more refined noncommutative multi-variate $m$-Noncrossing partitions presented by Novelli and Thibon in "Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra". "Planar diagrams" {referenced in item 10) contains both $[N^{(2)}]$ and $[A^{(2)}]$. (The earliest listing of the first few partition polynomials of $[A]$ and $[A^{(2)}]$ that I've seen is in a letter by Isaac Newton to Henry Oldenburg dated 1676.) Peter Bala in A108767 in 2008 stated a theorem very much related to these considerations, which amounts to stating that the substitution operation $$[N^{(2}]=[N][N] =[N]^2,$$ i.e., $$N_n^{(2)}(u_1,u_2,...,u_n) = N_n(N_1(u_1),N_2(u_1,u_2),...,N_n(u_1,...,u_n))$$ is equivalent to the aeration-deaeration operation $$[N^{(2)}]=([N][1,0,u_1,0,u_2,...])_{deaerate},$$ i.e., $$N_n^{(2)}(u_1,u_2,...,u_n) = N_{2n}(0,u_1,0,u_2,\cdots,0,u_n),$$ which generalizes to $$N^{(m+1)}_n(u_1,u_2,...,u_n) = N^{(m)}_n(N_1,N_2,...,N_n) = N_{mn}(0,0,...,u_1,0,0,...,u_2,\cdots,0,u_{mn})$$ with intervening swathes of $m$ zeros. Some implications of this algebra, which lift (if I understand N & T correctly) to the case of noncommutative indeterminates, are $$[N] = [A^{(m+1)}][A^{(m)}]$$ and $$[N^{(m)}] = [A^{(m)}][R]$$ and $$[N^{(m)}][R] = [A^{(m)}].$$ This algebra is a manifestation of the compositional inversion identity (CII), about $x=0$, $$ \left ( \frac{x^2}{((O^{(p-1)}(x))^{(-1)})_{dea}} \right )^{(-1)} |_{x \to x^{p}} = x^{p-1}\cdot (O^{(p)}(x))^{(-1)},$$ which translates into the partition polynomial substitution identity (PSI) $$[A][R]([A][O^{(p-1)}])_{dea}= [N]([A][O^{(p-1)}])_{dea}=[N] [A^{(p-1)}][O]$$ $$= ([A][O^{(p)}])_{dea} = [A^{(p)}][O],$$ where, e.g., $$(O^{(p)}(x))_{dea} = (x + c_1 x^{p+1} + c_2 x^{2p+1} + c_3 x^{3p+1}+\cdots)_{deaerate}$$ $$ = x + c_1 x^2 + c_2 x^3 + 3_3 x^4 + \cdots =O(x)$$ and $$[O^{(3)}]_{dea} =[1,0,0,c_1,0,0,c_2,...]_{dea}=[1,c_1,c_2,...] = [O].$$ The CII, and consequently the PSI, can be verified via the particular Lagrange-Schur (c. 1947)-Jabotinsky self-convolution expansion identity $$[x^n]\frac{\left ( \frac{F^{(-1)}(x)}{x} \right )^{k}}{k} = [x^n]\frac{\left ( \frac{x}{F(x)} \right )^{n+k}}{n+k}.$$

What are some other guises of the NCPs?

Tom Copeland
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    The lattice of noncrossing partitions is isomorphic to the interval $[e,c]$ of the absolute order on the symmetric group between the identity element $e$ and a fixed Coxeter element $c$. This way of viewing noncrossing partitions is key in generalizing the notion to other types. – Sam Hopkins Aug 21 '19 at 21:30
  • @SamHopkins, a ref paper? – Tom Copeland Aug 21 '19 at 21:37
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    This is by now very well known; a standard source is the monograph of Armstrong: https://arxiv.org/abs/math/0611106. – Sam Hopkins Aug 21 '19 at 21:41
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    In fact, what I said is also Lemma 4.3 in the first link you posted (the survey by Jon McCammond). – Sam Hopkins Aug 21 '19 at 21:46
  • See also "A bijection between noncrossing and nonnesting partitions for classical reflection groups" by B. Giraldo (senior thesis) – Tom Copeland Sep 09 '19 at 02:26
  • Sam Hopkins alludes to "Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups" by Drew Armstrong. – Tom Copeland Sep 09 '19 at 02:27
  • Other avatars in "Analytic combinatorics of non-crossing configurations" by Flajolet and Noy https://www.sciencedirect.com/science/article/pii/S0012365X98003720?via%3Dihub – Tom Copeland Sep 09 '19 at 02:50
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    Look up "Easy Quantum Groups" – JP McCarthy Sep 10 '19 at 20:55
  • @JPMcCarthy, a particularly easy ref? – Tom Copeland Sep 10 '19 at 22:15
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    See e.g. https://arxiv.org/pdf/1201.4723.pdf – JP McCarthy Sep 11 '19 at 08:52
  • See also "Noncrossing partitions and representations of quivers" by Ingalls and Thomas https://arxiv.org/abs/math/0612219 – Tom Copeland Sep 15 '19 at 22:16
  • Identification with non-decreasing parking functions and realtions to free probability and Hopf algebras in "Free cumulants, Schröder trees, and operads" by Josuat-Vergès, Menous, Novelli, and Thibon https://arxiv.org/abs/1604.04759 – Tom Copeland Sep 18 '19 at 17:44
  • Related "An instance of umbral methods in representation theory: the parking function module" by P. Petrullo and D. Senato https://arxiv.org/abs/0807.4840 – Tom Copeland Dec 09 '19 at 13:21
  • "Noncrossing partitions" by Rodica Simion (https://core.ac.uk/download/pdf/82503989.pdf) – Tom Copeland Jan 21 '20 at 23:03
  • For a connection to positroids, "Positive configuration space" by Nima Arkani-Hamed, Thomas Lam, Marcus Spradlin https://arxiv.org/abs/2003.03904 – Tom Copeland Apr 16 '20 at 01:36
  • "The category of noncrossing partitions" by Igusa https://arxiv.org/abs/1411.0196 – Tom Copeland Oct 20 '20 at 18:48
  • "Noncrossing partitions, Bruhat order and the cluster complex" by Philippe Biane and Matthieu Josuat-Vergès – Tom Copeland Mar 09 '22 at 17:24
  • "From parking functions to Gelfand pairs" by Kursar Aker and Mahir Bilen Can https://arxiv.org/abs/1002.1519 – Tom Copeland Jun 06 '23 at 05:18
  • "Counting partitions by genus. I. Genus 0 to 2" by Zuber (https://arxiv.org/abs/2303.05875). – Tom Copeland Sep 06 '23 at 11:57

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