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Statistics on monotonically ordered non-crossing partitions

Natasha Blitvic, Thomas Bray, Jacob Campbell, Alexandru Nica

TL;DR

This work analyzes monotone non-crossing partitions via two linked tree structures, $NC^{\mathrm{(mton)}}(n)$ and $NC^{\mathrm{(mton)}}_2(2n)$, to study block-counting statistics and the area statistic. It develops two recursive schemes (of the first and second kind) on the corresponding trees and leverages combinatorial Laplace transforms to obtain exact moment formulas and asymptotics. The main findings reveal a logarithmic growth regime for block-counting statistics and area in the monotone setting, contrasting with linear growth in the unordered case, and provide explicit expressions and asymptotics for outer/interval block counts and related quantities. The results connect to monotone cumulants and the monotone Poisson process, and they illustrate how these combinatorial tools yield precise, tractable recursions applicable to monotone probability theory.

Abstract

We study some combinatorial statistics defined on the set $NC^{(mton)}(n)$ of monotonically ordered non-crossing partitions of {1,...,n}, and on the set $NC_2^{(mton)}(2n)$ of monotonically ordered non-crossing pair-partitions of {1,...,2n}. Unlike in the analogous results known for unordered non-crossing partitions, the computations of expectations and variances for natural block-counting statistics on $NC^{(mton)}(n)$ and for the expectation of the area statistic on $NC_2^{(mton)}(2n)$ turn out to yield a logarithmic regime. An important role in our study is played by a nice tree structure on the disjoint union of the $NC^{(mton)}(n)$'s, which we use to streamline our arguments. As an illustration of how these ideas can be applied to calculations of cumulants in monotone probability, we discuss some combinatorial aspects of the monotonic Poisson process.

Statistics on monotonically ordered non-crossing partitions

TL;DR

This work analyzes monotone non-crossing partitions via two linked tree structures, and , to study block-counting statistics and the area statistic. It develops two recursive schemes (of the first and second kind) on the corresponding trees and leverages combinatorial Laplace transforms to obtain exact moment formulas and asymptotics. The main findings reveal a logarithmic growth regime for block-counting statistics and area in the monotone setting, contrasting with linear growth in the unordered case, and provide explicit expressions and asymptotics for outer/interval block counts and related quantities. The results connect to monotone cumulants and the monotone Poisson process, and they illustrate how these combinatorial tools yield precise, tractable recursions applicable to monotone probability theory.

Abstract

We study some combinatorial statistics defined on the set of monotonically ordered non-crossing partitions of {1,...,n}, and on the set of monotonically ordered non-crossing pair-partitions of {1,...,2n}. Unlike in the analogous results known for unordered non-crossing partitions, the computations of expectations and variances for natural block-counting statistics on and for the expectation of the area statistic on turn out to yield a logarithmic regime. An important role in our study is played by a nice tree structure on the disjoint union of the 's, which we use to streamline our arguments. As an illustration of how these ideas can be applied to calculations of cumulants in monotone probability, we discuss some combinatorial aspects of the monotonic Poisson process.

Paper Structure

This paper contains 21 sections, 20 theorems, 154 equations, 3 figures.

Table of Contents

  1. Introduction
  2. NC(n) and NCord(n). The NCord-tree structure.
  3. Block-counting statistics on NCord(n).
  4. The NCord2-tree, and some block-counting statistics on NCord2(2n).
  5. Another combinatorial statistic on NCord2(2n): area.
  6. Organization of the paper.
  7. The NCord-tree and some recursions related to it
  8. Description of the NCord-tree.
  9. The NCord-recursion of the first kind.
  10. The NCord-recursion of the second kind.
  11. Laplace transforms for the NCord-recursion of the first kind
  12. Statistics satisfying an NCord-recursion of the first kind
  13. Block-counting random variables -- proof of Theorem \ref{['thm:16']}.
  14. Block-counting tallied by size -- proof of Theorem \ref{['thm:17']}.
  15. An aside to Section \ref{['subsection:4-1']}: moments of the monotonic Poisson process
  16. ...and 6 more sections

Key Result

Theorem 1.6

Let the random variables $Y_n : NC^{\mathrm{(mton)}} (n) \to \mathbb{N}$ be as defined in (eqn:1-2b). $1^o$ One has $E[ \, Y_n \, ] = n - H_n + \frac{3}{2} - \frac{1}{n+1}, \ \ n \geq 2.$ As a consequence, it follows that where the meaning of "$\approx$" in (eqn:16b) is that the difference of the quantities on its two sides goes to $0$ for $n \to \infty$. $2^o$ One has $\mathrm{Var} [Y_n] = H_

Figures (3)

  • Figure 1: $\pi = \bigl\{ \, \{1,2\}, \, \{3,4,7,9\}, \, \{5,6\}, \, \{8\} \, \bigr\} \in NC(9)$ built in 4 steps, by inserting intervals. The building process gives a monotonic ordering $u$ of $\pi$: every block of $\pi$ retains the label assigned to it (in boldface font) at the moment when it was inserted in the picture, thus giving $u ( \, \{3,4,7,9\} \, ) = 1$, $u ( \, \{8\} \, ) = 2$, $u ( \, \{5,6\} \, ) = 3$, $u ( \, \{1,2\} \, ) = 4$.
  • Figure 2: $\pi = \bigl\{ \, \{1,2\}, \, \{3,4,7,9\}, \, \{5,6\}, \, \{8\} \, \bigr\} \in NC(9)$ built in 9 steps, by inserting points. The building process gives the path in the $NC^{\mathrm{(mton)}}$-tree which starts at the root of the tree and arrives at the monotonically ordered non-crossing partition from Figure 1.
  • Figure 3: $\pi = \bigl\{ \, \{1,6\}, \, \{2,3\}, \, \{4,5\}, \, \{7,8\}, \, \{9,14\}, \, \{10, 13\}, \, \{11,12\} \, \bigr\}$ in $NC_2 (14)$, and graph of the associated function $f_{\pi} : [0,14] \to [0, \infty )$.

Theorems & Definitions (55)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • Theorem 1.11
  • ...and 45 more