Partitions into distinct parts with bounded largest part
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RESEARCH
Partitions into distinct parts with bounded largest part Walter Bridges* * Correspondence:
[email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA
Abstract We prove an asymptotic formula for √ the number of partitions of n into distinct parts where the largest part is at most t n for fixed t ∈ R. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres’ asymptotic formula √ for distinct parts partitions when instead the number of parts is bounded by t n. Although equivalent to a circle method/saddle-point method calculation, the probabilistic approach motivates some of the more technical steps and even predicts the shape of the asymptotic formula, to some degree. Keywords: Partitions, Asymptotic analysis, Circle method, Probability Mathematics Subject Classification: 05A17, 11P82
1 Introduction A distinct parts partition λ of n is a set of positive integers {λ1 , . . . , λ } satisfying λ1 > λ2 > · · · > λ > 0;
λj = n.
j=1
Here, |λ| = n is the size of λ, and the λj are its parts. For example, the distinct parts partitions of 5 are 5, 4 + 1, and 3 + 2. Let d(n) denote the number of distinct parts partitions of n. These numbers are easily seen to be generated by the following infinite product d(n)xn = (1 + xk ). n≥0
k≥1
Pioneering work of Hardy and Ramanujan used the modular properties of the infinite product to obtain an asymptotic series for d(n) (and similar enumerations) after representing these coefficients as contour integrals around the origin ( [8], 7.1). The main term in Hardy and Ramanujan’s asymptotic series is π √ 1 √ n d(n) ∼ √ 3 e 3 . 4 4 3n 4
(1.1)
The circle method is now often used as an umbrella term for the asymptotic analysis of contour integrals, including Hardy–Ramanjuan’s method and its many variants, as well as certain cases of the saddle-point method. For an exposition of Hardy, Ramanujan, and
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W. Bridges Res. Number Theory (2020)6:40
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Rademacher’s original work, see [2], Ch. 5–6 and for the saddle-point method, see [7], Ch. VIII. A more recent approach to these asymptotic statistics, begun by Fristedt in [6] and used by Romik in [10], is to reformulate the proof using probability theory. This can make some of the steps more intuitive. We explain these ideas further in Sect. 2. Let t be a fixed positive real number. We study a restriction of d(n) defined as dt (n) := Coeff [xn ] Dt,n (x), where Dt,n (x) := (1 + xk ). √ k≤t n
√ Thus, dt (n) is the number of distinct parts partitions of n with largest part at most t n. The smallest possible largest part in a distinct parts partition of n with largest part at most √ t n is κ, where κ(κ − 1) κ(κ + 1) 2. We prove the following asymptotic formula for dt (n). Here and throughout, α denotes the greatest integer less than or equal to α and {α} := α − α . √ √ π 2, ∞ → −∞, √ Theorem 1 Let t > 2. Define β : implicitly as a function of t 2 3 so that t ue−βu 1= du.
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