arXiv subjects

Natasha Blitvić

Publications and source records attributed to Natasha Blitvić.

8 recordsLinked to original sources

arXiv2026

Colored Interlacing Triangles and Genocchi Medians

Colored interlacing triangles, introduced by Aggarwal-Borodin-Wheeler (2024), provide the combinatorial framework for the central limit theorem for probability measures arising from the Lascoux-Leclerc-Thibon (LLT) polynomials. Colored interlacing triangles depend on two key parameters: the number of colors $n$ and the depth of the triangle $N$. Recent work of Gaetz-Gao (2025) connects these objects to Schubert calculus and resolves the enumeration for $n=3$ and arbitrary depth $N$. However, the enumerative behavior for general $n$ has remained open. In this paper, we analyze the complementary regime: fixed depth $N=2$ and arbitrary number of colors $n$. We prove that in this setting, colored interlacing triangles considered up to a simultaneous relabeling of the colors are in bijection with Dumont derangements, identifying their enumeration with the Genocchi medians. This connects the probabilistic model to a rich hierarchy of classical combinatorial objects. In particular, under this bijection, the natural decomposition of a triangle into consecutive indecomposable blocks corresponds to the direct-sum decomposition of the associated permutation (readily yielding a generating function identity). Using a recent bijection of Poddar, Sawant, and Shankar (2026), we further identify these 'indecomposable' triangles with pure D-permutations. Furthermore, we introduce a $q$-deformation of this enumeration arising naturally from the LLT transition energy. For identity bottom row, we show that the corresponding statistic is the sum of the lower-crossing and lower-nesting statistics on Dumont derangements, and is additive over indecomposable triangles. This yields a new $q$-analogue of the median Genocchi numbers with an elegant continued-fraction expansion, as well as an interpretation of the indecomposable enumerators as Boolean cumulants of a family of probability measures on $\mathbb R$.

math.CO↗

arXiv2020

Permutations, moments, measures

Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, in the form of a continued fraction, for a fourteen-parameter family of such sequences and interpret these in terms of combinatorial statistics on the symmetric groups. Special cases include several classical and noncommutative probability laws, along with a substantial subset of the orthogonalizing measures in the q-Askey scheme, now given a new combinatorial interpretation in terms of elementary permutation statistics. This framework further captures a variety of interesting combinatorial sequences including, notably, the moment sequences associated to distributions of the numbers of occurrences of (classical and vincular) permutation patterns of length three. This connection between pattern avoidance and broader ideas in classical and noncommutative probability is among several intriguing new corollaries, which generalize and unify results previously appearing in the literature, while opening up new lines of inquiry. The fourteen combinatorial statistics further generalize to signed and colored permutations, and, as an infinite family of statistics, to the k-arrangements: permutations with k-colored fixed points, introduced here along with several related results and conjectures.

math.CO↗

arXiv2019

On a Generalized Fibonacci Recurrence

The generalized Fibonacci recurrence $g_n=g_{n-k}+g_{n-m}$ was recently used to demonstrate the theoretically optimal nature of limited senescence in morphologically symmetrically dividing bacteria. Here, we study this recurrence from a more abstract viewpoint, as a general model for asymmetric branching, and interpret solutions for different initial conditions in terms of branching-related quantities. We provide a compact diagrammatic representation for the evolution of this process which leads to an explicit binomial identity for the sums of elements lying on the diagonals $kx+my=n$ in Pascal's triangle $\mathbb N_0\times \mathbb N_0\ni(x,y)\mapsto {x+y\choose x}$, previously sought by Dickinson [Dic50], Raab [Raa63], and Green [Gre68].

math.CO↗

arXiv2019

Aging a little: The optimality of limited senescence in Escherichia coli

Recent studies have shown that even in the absence of extrinsic stress, the morphologically symmetrically dividing model bacteria Escherichia coli do not generate offspring with equal reproductive fitness. Instead, daughter cells exhibit asymmetric division times that converge to two distinct growth states. This represents a limited senescence / rejuvenation process derived from asymmetric division that is stable for hundreds of generations. It remains unclear why the bacteria do not continue the senescence beyond this asymptote. Although there are inherent fitness benefits for heterogeneity in population growth rates, the two growth equilibria are surprisingly similar, differing by a few percent. In this work we derive an explicit model for the growth of a bacterial population with two growth equilibria, based on a generalized Fibonacci recurrence, in order to quantify the fitness benefit of a limited senescence process and examine costs associated with asymmetry that could generate the observed behavior. We find that with simple saturating effects of asymmetric partitioning of subcellular components, two distinct but similar growth states may be optimal while providing evolutionarily significant fitness advantages.

q-bio.PE↗

arXiv2019

Type B Gaussian Statistics as Noncommutative Central Limits

We show that the noncommutative central limit theorem of Speicher can be adapted to produce the Gaussian statistics associated to Coxeter groups of type B, in the sense of Bożejko, Ejsmont, and Hasebe. Specifically, we show how type B Gaussian statistics naturally arise in systems of 'mixed spins', providing a new application of Speicher's argument and paving the way for the transfer of known results from the bosonic/fermionic settings to such broader contexts.

math.PR↗

arXiv2012

Two-parameter Non-commutative Central Limit Theorem

The non-commutative Central Limit Theorem (CLT) introduced by Speicher in 1992 states that given almost any sequence of non-commutative random variables that commute or anti-commute pair-wise, the *-moments of the normalized partial sum S_N=(b_1+...+ b_N)/\sqrt{N} are given by a Wick-type formula refined to count the number of crossings in the underlying pair-partitions. When coupled with explicit matrix models, the theorem yields random matrix models for creation and annihilation operators on the q-Fock space of Bozejko and Speicher. In this paper, we derive a non-commutative CLT when the pair-wise commutation coefficients are real numbers (as opposed to signs). The statistics of the limiting random variable are a second-parameter refinement of those above, jointly indexing the number of crossings and nestings in the underlying pair-partitions. Coupled with analogous matrix constructions, the theorem yields random matrix models for creation and annihilation operators on the recently introduced (q,t)-Fock space.

math.PR↗

arXiv2012

The $(q,t)$-Gaussian Process

We introduce a two-parameter deformation of the classical Bosonic, Fermionic, and Boltzmann Fock spaces that is a refinement of the $q$-Fock space of [BS91]. Starting with a real, separable Hilbert space $H$, we construct the $(q,t)$-Fock space and the corresponding creation and annihilation operators, $\{a_{q,t}(h)^\ast\}_{h\in H}$ and $\{a_{q,t}(h)\}_{h\in H}$, satifying the $(q,t)$-commutation relation $a_{q,t}(f)a_{q,t}(g)^\ast-q \,a_{q,t}(g)^\ast a_{q,t}(f)= _{_H}\, t^{N},$ for $h,g\in H$, with $N$ denoting the number operator. Interpreting the bounded linear operators on the $(q,t)$-Fock space as non-commutative random variables, the analogue of the Gaussian random variable is given by the deformed field operator $s_{q,t}(h):=a_{q,t}(h)+a_{q,t}(h)^\ast$, for $h\in H$. The resulting refinement is particularly natural, as the moments of $s_{q,t}(h)$ are encoded by the joint statistics of crossings \emph{and nestings} in pair partitions. Furthermore, the orthogonal polynomial sequence associated with the normalized $(q,t)$-Gaussian $s_{q,t}$ is that of the $(q,t)$-Hermite orthogonal polynomials, a deformation of the $q$-Hermite sequence that is given by the recurrence $zH_n(z;q,t)=H_{n+1}(z;q,t)+[n]_{q,t}H_{n-1}(z;q,t),$ with $H_0(z;q,t)=1$, $H_1(z;q,t)=z$, and $[n]_{q,t}=\sum_{i=1}^n q^{i-1}t^{n-i}$. The $q=0<t$ specialization yields a new single-parameter deformation of the full Boltzmann Fock space of free probability. The probability measure associated with the corresponding deformed semicircular operator turns out to be encoded, in various forms, via the Rogers-Ramanujan continued fraction, the Rogers-Ramanujan identities, the $t$-Airy function, the $t$-Catalan numbers of Carlitz-Riordan, and the first-order statistics of the reduced Wigner process.

math.OA↗

arXiv2011

On the norm of the $q$-circular operator

The $q$-commutation relations, formulated in the setting of the $q$-Fock space of Bożjeko and Speicher, interpolate between the classical commutation relations (CCR) and the classical anti-commutation relations (CAR) defined on the classical bosonic and fermionic Fock spaces, respectively. Interpreting the $q$-Fock space as an algebra of "random variables" exhibiting a specific commutativity structure, one can construct the so-called $q$-semicircular and $q$-circular operators acting as $q$-deformations of the classical Gaussian and complex Gaussian random variables, respectively. While the $q$-semicircular operator is generally well understood, many basic properties of the $q$-circular operator (in particular, a tractable expression for its norm) remain elusive. Inspired by the combinatorial approach to free probability, we revist the combinatorial formulations of these operators. We point out that a finite alternating-sum expression for $2n$-norm of the $q$-semicircular is available via generating functions of chord-crossing diagrams developed by Touchard in the 1950s and distilled by Riordan in 1974. Extending these norms as a function in $q$ onto the complex unit ball and taking the $n\to\infty$ limit, we recover the familiar expression for the norm of the $q$-semicircular and show that the convergence is uniform on the compact subsets of the unit ball. In contrast, the $2n$-norms of the $q$-circular are encoded by chord-crossing diagrams that are parity-reversing, which have not yet been characterized in the combinatorial literature. We derive certain combinatorial properties of these objects, including closed-form expressions for the number of such diagrams of any size with up to eleven crossings. These properties enable us to conclude that the $2n$-norms of the $q$-circular operator are significantly less well behaved than those of the $q$-semicircular operator.

math-ph↗