[Paper Review] Asymptotics of Young tableaux in the strip, the $d$-sums

Amitai Regev|arXiv (Cornell University)|Apr 26, 2010

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper revisits the asymptotic analysis of $d$-sums of standard Young tableaux in the strip, correcting a previously reported constant factor in the asymptotic formula for $T_{d,ds}^{(eta)}(dm)$ when $d > 1$. Using refined applications of the Young-Frobenius formula and Laplace's method, the authors derive a corrected asymptotic expression that aligns with direct computations and special cases, replacing Corollary 4.4 of [5] for $d > 1$. The key contribution is a precise, validated asymptotic formula involving gamma functions, powers of $m$, and combinatorial factors tied to $d$, $s$, and $\alpha$.

ABSTRACT

The asymptotics of the "strip" sums $S_\ell^{(\al)}(n)$ and of their $d$-sums generalizations $T_{d,ds}^{(\al)}(dm)$ (see Definition~ ef{definition1}) were calculated in~\cite{regev}. It was recently noticed that when $d>1$ there is a certain confusion about the relevant notations in~\cite{regev}, and the constant in the asymptotics of these $d$-sums $T_{d,ds}^{(\al)}(dm)$ seems to be off by a certain factor. Based on the techniques of~\cite{regev} we again calculate the asymptotics of the $d$-sums $T_{d,ds}^{(\al)}(dm)$. We do it here carefully and with complete details. This leads to Theorem~ ef{d.sum222} below, which replaces Corollary 4.4 of~\cite{regev} in the cases $d>1$.

Motivation & Objective

• To correct the asymptotic formula for $T_{d,ds}^{(eta)}(dm)$, which was found to contain an incorrect constant factor in [5] when $d > 1$.
• To re-derive the asymptotics of $d$-sums $T_{d,ds}^{(eta)}(dm)$ using rigorous and complete analytical techniques based on the Young-Frobenius formula.
• To validate the new asymptotic formula through direct computation in special cases where closed forms exist, including $s=1$, $d=1$, and $s=d=2$, $\alpha=1$.
• To ensure consistency with known results such as Catalan numbers and hypergeometric identities, particularly for $T_{2,4}^{(1)}(2m)$.
• To provide a corrected and complete asymptotic expression that generalizes the earlier result in [5] for $d > 1$.

Proposed method

• Applies the Young-Frobenius formula for $f^\lambda$, the number of standard Young tableaux of shape $\lambda$, and expands it asymptotically using Stirling's approximation.
• Parametrizes the parts of $\lambda$ as $\lambda_i = m/s + b_i\sqrt{m}$, treating $b_i$ as bounded variables to model fluctuations around the mean size.
• Uses the Vandermonde-type determinant $D_s(b_1, \dots, b_s)$ to capture the relative differences in row lengths and their contribution to $f^\lambda$.
• Applies Laplace's method to the resulting integral over the $b_i$'s, identifying the dominant contribution from the Gaussian-like term $e^{-(ds/2)(b_1^2 + \cdots + b_s^2)}$.
• Integrates the asymptotic expression for $f^\lambda$ raised to the power $\alpha$, leading to a multivariate Gaussian integral that evaluates to a product of gamma functions.
• Derives the final asymptotic formula by combining the leading-order term from the tableaux count with the normalization and scaling factors arising from the integral and the $d$-symmetry in the partition structure.

Experimental results

Research questions

• RQ1 Why was the constant factor in the asymptotic formula for $T_{d,ds}^{(\alpha)}(dm)$ in [5] incorrect for $d > 1$, and what is the correct factor?
• RQ2 How do the asymptotics of $d$-sums of standard Young tableaux behave when $d > 1$, particularly in the strip $\ell(\lambda) \leq ds$?
• RQ3 Can the corrected asymptotic formula be validated against known closed-form expressions in special cases such as $s=1$, $d=1$, or $s=d=2$, $\alpha=1$?
• RQ4 Does the new asymptotic formula for $T_{d,ds}^{(\alpha)}(dm)$ agree with direct asymptotic evaluation via Stirling's formula in cases like the Catalan numbers?
• RQ5 What is the precise role of the $d$-symmetry in the partition structure $\lambda = (\mu_1^d, \mu_2^d, \dots)$ in shaping the asymptotic behavior of $f^\lambda$?

Key findings

• The corrected asymptotic formula for $T_{d,ds}^{(\alpha)}(dm)$ replaces Corollary 4.4 of [5] for $d > 1$, with a new constant factor derived from a complete re-analysis of the Laplace method application.
• For $s=1$, the formula reduces correctly to a single tableau $\lambda = (m^d)$, and the product of correction factors equals 1, confirming consistency.
• When $d=1$, the formula reduces to the original result in [5], confirming correctness in the $d=1$ case and validating the generalization.
• For $d=1$, $\alpha=1$, the formula matches the known asymptotic for the sum of $f^\lambda$ over partitions with at most $s$ rows, consistent with [5, (F.4.5.1)].
• For $d=1$, $\alpha=2$, the formula reproduces the asymptotic for the sum of squares of $f^\lambda$, matching the known asymptotic for the central binomial coefficient and Catalan numbers.
• For $s=d=2$, $\alpha=1$, the formula matches the asymptotic of sequence A005700, which counts certain lattice paths, via direct Stirling approximation of the closed-form expression.