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Bounds for the number of multidimensional partitions
Combinatorics · Submitted 25 Mar 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $\log p_d(n)$ is up to an absolute constant equal to $n^{1-1/d}$.
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The abstract, sentence by sentence
We obtain estimates for the number $p_d(n)$ of $(d-1)$-dimensional integer partitions of a number $n$. It is known that the two-sided inequality $C_1(d)n^{1-1/d}<\log p_d(n)< C_2(d)n^{1-1/d}$ is always true and that $C_1(d)>1$ whenever $\log n> 3d$. However, establishing the $``$right$"$ dependence of $C_2$ on $d$ remained an open problem. We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $\log p_d(n)$ is up to an absolute constant equal to $n^{1-1/d}$. Besides, we provide estimates of $p_d(n)$ for different ranges of $d$ in terms of $n$, which give the asymptotics of $\log p_d(n)$ in each case.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We obtain estimates for the number $p_d(n)$ of $(d-1)$-dimensional integer partitions of a number $n$. | contribution 0.89 | 0.95 | 0.10 | asserted 1.00 |
| s1 | It is known that the two-sided inequality $C_1(d)n^{1-1/d}<\log p_d(n)< C_2(d)n^{1-1/d}$ is always true and that $C_1(d)>1$ whenever $\log n> 3d$. | background 1.00 | 0.05 | 0.05 | asserted 1.00 |
| s2 | However, establishing the $``$right$"$ dependence of $C_2$ on $d$ remained an open problem. | gap 1.00 | 0.04 | 0.03 | asserted 0.99 |
| s3 | We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $\log p_d(n)$ is up to an absolute constant equal to $n^{1-1/d}$. | result 0.96 | 0.96 | 0.74 | asserted 1.00 |
| s4 | Besides, we provide estimates of $p_d(n)$ for different ranges of $d$ in terms of $n$, which give the asymptotics of $\log p_d(n)$ in each case. | result 0.59 | 0.95 | 0.16 | asserted 1.00 |
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Original abstract
We obtain estimates for the number $p_d(n)$ of $(d-1)$-dimensional integer partitions of a number $n$. It is known that the two-sided inequality $C_1(d)n^{1-1/d}<\log p_d(n)< C_2(d)n^{1-1/d}$ is always true and that $C_1(d)>1$ whenever $\log n> 3d$. However, establishing the $``$right$"$ dependence of $C_2$ on $d$ remained an open problem. We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $\log p_d(n)$ is up to an absolute constant equal to $n^{1-1/d}$. Besides, we provide estimates of $p_d(n)$ for different ranges of $d$ in terms of $n$, which give the asymptotics of $\log p_d(n)$ in each case.
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