Monday, July 27, 2026

From Partitions to Pobability Distributions

From Partitions to Probability

Author: E A Thomas
Date: 27 July 2026

Abstract

Classical parametric discrete distributions are derived from limiting processes or stochastic assumptions. In contrast, combinatorial number theory generates highly structured frequency sequences—such as the partition step pattern —whose internal symmetries are rarely viewed through a probabilistic lens. This paper establishes a paradigm shift by embedding into a Cartesian probability framework, revealing it as a strictly unimodal distribution with well-defined moments and asymptotic normality. We then generalize this construction to the General Sequence-Difference Distribution (GSDD) , defined by cumulative summation of finite integer sequences and . We prove that the GSDD is universal for all bounded unimodal discrete distributions (exactly for rational data, and arbitrarily closely for real data). Crucially, by specializing and to specific mathematical sequences—arithmetic, geometric, logarithmic, square-root, hyperbolic, Fibonacci, and primes—we construct a tunable "vocabulary" of distributional shapes with controlled tail-heaviness, skewness, and kurtosis. This bridges pure combinatorics with applied statistics, offering a constructive, interpretable alternative to conventional discrete models.

1. Introduction

The modelling of discrete count data is a cornerstone of statistics. Families such as the Poisson, Binomial, and Negative Binomial arise from natural stochastic processes (rare events, Bernoulli trials, waiting times). When data exhibit over-dispersion or complex tail behaviour, practitioners turn to the Beta-Binomial or the discrete Pearson system—families defined by differential equations or moment-matching heuristics. While effective, these models often obscure the underlying combinatorial structure of the data.

Simultaneously, the field of partition theory—the study of writing integers as sums of positive integers—generates deeply structured sequences. The total partition count grows exponentially, but the internal step pattern , where counts partitions with largest part exactly , exhibits remarkable regularity: strict unimodality, approximate symmetry, and known asymptotic normality.¹

Historically, has been treated exclusively as a frequency distribution. However, if we place on a Cartesian grid—with part-size on the x-axis and normalized count on the y-axis—we obtain a proper probability mass function. This observation is not a trivial relabeling; it equips the sequence with expectation, variance, and entropy, allowing us to compare partitions of different integers using statistical metrics.

This paper pursues this observation to its logical extreme. We ask: What if the combinatorial structure of integer sequences is the fundamental driver of probability shapes, rather than differential equations?

We answer this by introducing the General Sequence-Difference Distribution (GSDD) . Given two arbitrary finite positive integer sequences and with equal sum, we form a difference sequence . The cumulative sum of yields a unimodal integer sequence; normalizing this sequence produces a probability distribution. We prove a Universality Theorem demonstrating that this single construction contains every strictly unimodal discrete distribution.

Crucially, beyond mere universality, the framework provides granular, interpretable control. We systematically explore sub-families of the GSDD defined by the mathematical nature of and :

  • Arithmetic sequences yield uniform and triangular distributions.
  • Geometric sequences produce exponential heavy tails.
  • Logarithmic, Square-root, and Hyperbolic sequences generate extreme skew, flattened domes, and winner-take-all concentration, respectively.
  • Fibonacci and Prime sequences introduce recursive or erratic curvature.

The paper is structured as follows. Section 2 introduces the partition step pattern , first through an intuitive decomposition, then via a formal constructive algorithm, and establishes the Cartesian probability framework. Section 3 derives the foundational statistical properties of partition-derived distributions. Section 4 presents the GSDD construction. Section 5 proves the Universality Theorem. Section 6 presents the spectrum of sub-families. Section 7 derives closed-form moments for general and special cases. Section 8 concludes. An Annex provides a step-by-step Monte Carlo simulation and fitting demonstration.

2. Partitions and the Step Pattern

2.1 Defining the Partition Table

A partition of a positive integer is a non-increasing sequence of positive integers summing to .² For example, the partitions of 5 are:
.

Let denote the total number of partitions. Let denote the number of partitions of with largest part exactly , for .

By the well-known conjugation symmetry of partitions (Ferrers diagrams), is also the number of partitions of with exactly parts.³ The step pattern is the finite sequence of these values:


2.2 The Step Pattern : A Worked Example

To ground the theory, we present the partition table for . The sequence is:

(Part Size)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

1

10

33

64

84

90

82

70

54

42

30

22

15

11

7

5

3

2

1

1

Summation check: .

Observe the strict unimodality: the sequence rises strictly to a peak at (frequency 90), and then descends strictly to (frequency 1).

2.3 The Conventional Approach: The Generating Function

The partition numbers are classically obtained as coefficients of the infinite product:⁴


This is a compact and elegant theoretical expression, central to analytic number theory. However, extracting the step pattern from this product requires either symbolic manipulation of power series or advanced asymptotic analysis—both of which obscure the simple combinatorial structure underlying the sequence. While the generating function is mathematically superior for proving theorems, it is less suited for the constructive, algorithmic aims of this paper.

2.4 An Intuitive Decomposition of the Step Pattern

Before introducing formal algorithms, we observe a fundamental symmetry that makes the step pattern accessible by direct construction.

Consider the partitions of . We can construct them recursively by leading digit. Given a leading part , the remainder is any partition of using parts no larger than . This yields the following decomposition:

Leading part

Partitions beginning with

Count

1

1

2

, , , ,

5

3

, , , , , , ,

8

4

, , , , , , , ,

9

5

, , , , , ,

7

6

, , , ,

5

7

, ,

3

8

,

2

9

1

10

1

Thus:


Why this decomposition matters: This constructive approach—building partitions by leading digit—is the natural complement to the formal DP recurrence. It reveals that the step pattern is not merely a frequency list but a recursive object with deep internal structure.⁵

2.5 A Formal Constructive Algorithm (Dynamic Programming)

While the intuitive decomposition above is excellent for understanding and hand computation, a formal algorithm is required for systematic generation of for arbitrary . We now introduce the dynamic programming recurrence that computes the entire table using only integer addition.

Let denote the number of partitions of integer using parts no larger than (i.e., largest part ). The following recurrence computes the entire table using only integer addition:


with base cases:

  • for all (the empty partition).
  • for all .

Interpretation:

  • : Partitions of that do not use the part of size .
  • : Partitions of that use at least one part of size (remove that , and the remainder is a partition of with parts still ).

Once the DP table is computed, the step pattern is extracted by the simple formula:


Why this works: If the largest part is exactly , then removing that one part leaves a partition of whose parts are all . The number of such partitions is exactly .

2.6 Worked Example: Computing with the DP

We compute the DP table up to . Here is the relevant portion:

...

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

1

2

2

2

2

2

2

2

3

1

2

3

3

3

3

3

3

4

1

3

4

5

5

5

5

5

5

1

3

5

6

7

7

7

7

6

1

4

7

9

10

11

11

11

7

1

4

8

11

13

14

15

15

8

1

5

10

15

18

20

21

22

9

1

5

12

18

23

26

28

30

10

1

6

14

23

30

35

38

42

Now extract using :

1

9

1

2

8

5

3

7

8

4

6

9

5

5

7

6

4

5

7

3

3

8

2

2

9

1

1

10

0

1

Thus:


This matches the intuitive decomposition exactly. The DP algorithm formalizes the recursive structure that the constructive method revealed.

2.7 Why This Algorithm Matters

This DP recurrence is the constructive heart of the paper. It establishes that:

  1. is computable in time using only integer arithmetic.
  2. The computation is transparent: each cell has a direct combinatorial interpretation.
  3. The extraction directly links the step pattern to the finite-difference construction that follows.

The generating function remains valuable as a theoretical tool, but the DP algorithm—and the constructive decomposition that precedes it—are the practical bridges between partition theory and the GSDD. They align with the constructive philosophy of the paper: we build distributions from integer sequences, not from differential equations or asymptotic expansions.

3. The Cartesian Framework and Probabilistic Interpretation

3.1 From Frequency to Probability

In number theory, is merely a vector of frequencies. To reinterpret it as a probability distribution, we embed it in the standard Cartesian probability framework:

  • Domain (X-axis): The index , representing the size of the largest part.
  • Mass (Y-axis): The normalized frequency .

Because the total sum of the frequencies is , normalization yields a proper PMF:


3.2 Why This Is Not Trivial

Treating as a probability distribution equips it with statistical moments. The mean part size is . The variance measures the spread of partition sizes around this mean. This probabilistic lens allows us to compare partitions of different integers using standard statistical metrics, rather than purely combinatorial growth rates.

For , the normalized distribution yields:

  • Mean ():
  • Variance ():
  • Standard Deviation:

This quantitative summary tells us that, for a random partition of 20 selected uniformly, the expected largest part is approximately 7.38, with a standard deviation of about 3.15 parts.

4. Statistical Properties of Partition Distributions

Analyzing for varying reveals intrinsic statistical laws:

  1. Unimodality: For all , is strictly unimodal. The sequence rises to a single peak and then declines monotonically. The location of the peak (the mode) grows approximately as in the asymptotic limit.⁶
  2. Asymptotic Behavior: The Gumbel Tail vs. the Normal Bulk. Let be a random variable with probability mass function . This represents the largest part of a uniformly random partition of . Two distinct asymptotic regimes emerge:
    • The Extreme Right Tail (Gumbel): Erdős and Lehner (1941) proved that the standardized largest part converges in distribution to the Gumbel (extreme value) distribution. Specifically, there exist centering and scaling constants and such that:


This governs the rare, large fluctuations of the maximum part far to the right of the peak.⁷

    • The Central Bulk (Gaussian): In contrast, the local behavior of the probability mass function near its center (the mode) converges to a Gaussian (normal) curve. This follows from saddle-point approximations to the partition generating function , which reveal that the coefficients are locally normally distributed around the mean.⁸
    • Reconciling the Two: The Gumbel distribution describes the extreme rightward fluctuations. The normal curve describes the high-probability "bulk" of the distribution. For finite , the overall shape is a slightly right-skewed bell-curve; the skewness diminishes as , and the bulk becomes increasingly symmetric and normal.
  1. Skewness (Clarified): For small , the distribution exhibits positive skew (right-skewed) in the standard statistical sense. This means the right tail is longer and thinner than the left tail, which pulls the mean to the right of the peak. We acknowledge that this convention can appear counterintuitive—visually, the bulk of the mass lies on the left side of the support—but the formal definition of skewness is determined by the direction of the stretched tail. As increases, the skewness tends toward zero, and the distribution becomes more symmetric.
  2. Discrete Support: Unlike continuous normals or Gumbel distributions, the support of strictly grows with (from to ). Therefore, the family does not converge to a single fixed distribution; instead, it represents a sequence of expanding distributions whose shape (in the bulk) tends toward normal, while the extreme tail follows a Gumbel law.

5. The General Sequence-Difference Distribution (GSDD)

5.1 The General Construction

Let and be any finite sequences of positive integers such that .

The GSDD is constructed by a simple finite-difference process:

  1. Form the difference sequence:


of length .

  1. Take the cumulative sum:


(Note that , so the support ends at .)

  1. Normalize:


The resulting distribution is strictly unimodal with peak at index , provided all entries of and are positive.

Interpretation: The sequence controls the rise from zero to the peak; the sequence controls the fall from the peak back to zero. The construction is entirely natural: we build the distribution by climbing up (via ) and then climbing down (via ).

5.2 The Partition Distribution as a Special Case

The step pattern is recovered precisely when the cumulative sum equals the raw frequencies . In this case:

  • The left sequence is the finite differences of the left side of , rising from zero to the peak.
  • The right sequence is the positive magnitudes of the finite differences on the right side, falling from the peak back to zero.

The sequences and are defined directly from the left and right increments, without any reversal. For , for example:


and the cumulative sum of yields exactly .

5.3 Skew Symmetry

The construction reveals a natural symmetry. If we swap the roles of and —i.e., construct a new distribution from and —the resulting distribution is the mirror image of the original, reflected about the peak.

More precisely, the distribution generated by has the same shape but reversed support order. Thus, by choosing which sequence is assigned to and which to , the practitioner controls the direction of the skew. This orientation symmetry is a direct consequence of the natural construction .

6. The Universality Theorem and Its Proof

We now establish the central theoretical result. While the construction appears intuitive, the proof requires careful attention to integrality and strict monotonicity.

6.1 Exact Representation for Rational, Strictly Unimodal Distributions

Definition (Strict Unimodality): A discrete distribution with is strictly unimodal if there exists an index such that:


Theorem 1 (Exact Universality over Rationals):
Let be a strictly unimodal discrete distribution with a peak at index . If every is a rational number, then there exists a GSDD with integer sequences and whose normalized cumulative sums exactly equal .

Proof:

  1. Clearing Denominators: Since is a probability distribution, its values sum to 1. However, the GSDD constructs raw integer heights before normalization. Let be the least common multiple of the denominators of all . Define the integer-scaled heights . The peak value is .
  2. Constructing the Left Sequence :
    Define the finite difference of the left side, rising from zero:


Because the left side is strictly increasing, every element of is a positive integer. Taking the telescoping sum:


  1. Constructing the Right Sequence :
    Define the right decrement sequence, falling to zero:


Because the right side is strictly decreasing, every element of is a positive integer. Taking the telescoping sum:


  1. Telescoping the Cumulative Sum:
    Form the difference sequence     and compute the cumulative sums .
    • For , the cumulative sum successively adds the left differences, which telescopes perfectly to .
    • For , the cumulative sum subtracts the right decrements, which telescopes perfectly to .
    • At , the cumulative sum becomes , giving the exact support length.
  2. Normalization: Let . Therefore:


Thus, the constructed GSDD exactly reproduces .

6.2 Corollary: Approximation of Real-Valued (Irrational) Distributions

Corollary 1: For any strictly unimodal distribution with real values , and for any tolerance , there exists a GSDD distribution such that the total variation distance .

Proof:
Let . Choose a large integer . Define (adding 1 ensures strict positivity and strict unimodality is preserved). By Theorem 1, there exists a GSDD for the rational distribution . The rounding error per coordinate is at most , so the summed absolute error is at most .

6.3 Corollary: Handling Plateaus (Non-Strict Unimodality)

Corollary 2: If a target distribution has a plateau at the peak (i.e., ), the GSDD can represent it exactly by allowing non-negative integers in sequences and (permitting zeros in the difference sequences). Alternatively, it can be represented as an equal mixture of two strictly unimodal GSDDs with peaks at and .

7. The Spectrum of Sub-Families from Specific Mathematical Sequences

Having established universality, we now leverage the flexibility of sequences and . By selecting specific mathematical sequences, we gain precise, interpretable control over the shape of the distribution.

7.1 The Arithmetic Sub-Family (Uniform/Triangular)

Let and for symmetry, or simply .

  • Difference Behavior: Constant increments.
  • Distributional Effect: Produces a discrete triangular distribution (if symmetric) or linear tails.
  • Special Case: If , the cumulative sum yields a perfectly uniform distribution over the support.

7.2 The Geometric Sub-Family (Heavy-Tailed)

Let and scaled to match the sum.

  • Difference Behavior: Exponential growth.
  • Distributional Effect: Cumulative sums produce exponential tails, suitable for modeling over-dispersed count data (e.g., wealth distribution).

7.3 The Square-Root Sub-Family (Controlled Flattening)

Let .

  • Difference Behavior: Differences shrink monotonically as .
  • Distributional Effect: When is reversed, early differences are large, followed by tiny increments near the peak, creating a broad, flattened dome (platykurtic).

7.4 The Logarithmic Sub-Family (Extreme Skew)

Let .

  • Difference Behavior: Differences shrink rapidly as .
  • Distributional Effect: The cumulative sum rises steeply then virtually stalls, producing extreme positive skew (if on the left) or negative skew. Ideal for "winner-take-all" phenomena.

7.5 The Hyperbolic (Harmonic) Sub-Family (Extreme Concentration)

Let for , where is a large scaling constant to ensure integer positivity.

  • Difference Behavior: The successive differences shrink quadratically: .
  • Distributional Effect: When reversed and cumulatively summed, the distribution rises almost instantly to its peak and then crawls upward in infinitesimal increments. The first support point often carries an overwhelming majority of the total probability mass (>50%). This produces an exceptionally long, thin right tail that decays extremely slowly.
  • Asymptotics: , so the total mass scales logarithmically, but the distribution of that mass is entirely front-loaded.
  • Applications: Citation counts (a tiny fraction of papers receive nearly all citations), city population distributions (a single megacity dominates), and word frequencies in natural language at the extreme end of Zipf's law.

7.6 The Fibonacci Sub-Family (Recurrent Curvature)

Let where are Fibonacci numbers.

  • Difference Behavior: Recurrent growth.
  • Distributional Effect: Introduces subtle inflection points or "shoulders" in the PMF.

7.7 The Prime Sub-Family (Irregular / Erratic)

Let and be sequences of consecutive primes.

  • Difference Behavior: Erratic, non-monotonic jumps.
  • Distributional Effect: Produces jagged, highly irregular distributions, demonstrating the GSDD's robustness as a non-parametric interpolator.

7.8 Comparison Table of Curvature Control

Sequence Type

Increment Behavior

Tail Shape

Concentration

Ideal Use Case

Arithmetic

Constant

Light, bounded

Uniform spread

Surveys, bounded scores

Geometric

Exponential growth

Heavy exponential

Moderate spread

Wealth, income brackets

Square-Root

Diminishing (power 0.5)

Flattened shoulders

Moderate concentration

Consensus data, quality control

Logarithmic

Diminishing (power 1)

Very heavy, slow decay

High concentration

City sizes, species abundance

Hyperbolic (1/n)

Rapidly diminishing (power 2)

Extremely heavy, ultra-slow decay

Extreme concentration

Citation counts, word frequencies

Fibonacci

Recurrent

Medium with inflection

Variable

Biological growth stages

Primes

Erratic

Irregular

Unstable

Cryptographic / benchmark testing

8. Closed-Form Normalization and Moments

While the GSDD is computationally straightforward (cumulative sums and normalization), deriving closed-form expressions provides theoretical insight and practical efficiency.

Let , , with , and total difference length . The support is .

Theorem 2 (Normalization Constant):
The normalizing constant is given by:


Proof:
We exploit the identity . Substituting and changing indices yields the result.

Theorem 3 (First Moment):
The raw first moment is given by:


Consequently, the mean is .

Proof:
Using , substituting , and simplifying, yields the theorem.

Higher moments follow from using Faulhaber's formulas.

8.1 Specialization: Arithmetic Sub-Family (Symmetric)

Let . Applying Theorem 2:


Due to symmetry, the mean is exactly the midpoint:


8.2 Specialization: Logarithmic and Hyperbolic Sub-Families

For , Theorems 2 and 3 provide immediate computation. Asymptotically, , leading to:


which approaches for large , confirming the extreme concentration of mass at small indices.

For the hyperbolic case , the sum scales as as well, but the distribution of this mass is radically different. The cumulative sum reaches nearly its maximum value within the first two or three indices. The mean asymptotically approaches a constant (close to 1) rather than growing with , capturing the "winner-take-all" phenomenon.

9. Conclusion and Future Work

We have established a complete theoretical framework that transforms the classical study of integer sequences into a rigorous, flexible theory of discrete probability distributions.

Summary of Contributions:

  1. Foundational Bridge: We began with the partition step pattern , reinterpreting its normalized frequencies as a bona fide probability distribution on the integer lattice.
  2. Intuitive Decomposition: We introduced a natural constructive method—building partitions by leading digit—which reveals the recursive structure of the step pattern and provides an accessible entry point.
  3. Constructive Algorithm: We formalized the DP recurrence as a transparent, arithmetic method for computing , linking partitions directly to the GSDD construction.
  4. Universal Family: We introduced the GSDD, defined purely by finite differences. Theorem 1 proves exact representation of all strictly unimodal rational distributions, with Corollary 1 extending to real data via approximation.
  5. Closed-Form Moment Theory: Theorems 2 and 3 provide analytic tools to compute and directly from the raw sequences.
  6. Tunable Spectrum: By specializing to Arithmetic, Geometric, Logarithmic, Square-root, Hyperbolic, Fibonacci, and Prime sequences, we constructed a "vocabulary" of distributional shapes with interpretable tail and curvature properties.

Constructive Paradigm: This work proposes a paradigm shift. Instead of selecting a parametric family and estimating its parameters, the practitioner constructs the distribution by selecting the integer sequences and . This mirrors Pearson's mechanical models but translates them into the discrete, combinatorial realm.⁹

Future Work:

  • Maximum Likelihood Estimation (MLE): Formulating the likelihood over the integer lattice and developing dynamic programming algorithms for sequence recovery.
  • Multivariate Extensions: Generalizing to 2D sequences (plane partitions) for bivariate discrete distributions.
  • Continuous Limits: Investigating the scaling limits of the GSDD as , connecting to stable distributions and Lévy processes.
  • Bayesian Inference: Placing partition priors over the space of sequences for non-parametric Bayesian modeling.

Annex A: Monte Carlo Simulation and Fitting Procedure

This annex demonstrates the complete lifecycle of a GSDD model: generating random samples, and fitting the exact GSDD to recover the original sequences.

A.1 Monte Carlo Generation (Sampling)

Let a GSDD be defined by sequences and , resulting in the normalized PMF for .

To draw a random sample of size from this distribution via the inverse transform method:

  1. Compute the CDF: Construct and for .
  2. Inverse Transform: For each draw :
    • Generate .
    • Find the smallest such that . This is the sampled value.
  3. Result: The output is an integer-valued sample .

A.2 Fitting the GSDD to Empirical Data (Reconstruction)

Suppose we have an empirical dataset drawn from an unknown unimodal distribution with support .

  1. Compute Empirical Frequencies: Let be the raw count of observations equal to . The empirical PMF is . Ensure strict unimodality (smooth if necessary).
  2. Scale to Integers: Multiply by a large integer (e.g., ) to obtain integer heights .
  3. Extract Left Sequence : The positive jumps on the left side are:


  1. Extract Right Sequence : The decrements on the right side are:


  1. Validate: Reconstruct the GSDD from and . By Theorem 1, the normalized cumulative sums equal , which is the fitted PMF.

A.3 Recovery Test

Generate from a known source (e.g., Hyperbolic sub-family). Draw samples. Apply the fitting algorithm. As , the recovered and converge to the source sequences by the law of large numbers. This validates the framework without requiring numerical optimization.

End of Manuscript

Footnotes

¹ The asymptotic normality of the step pattern in the bulk is a consequence of the central limit theorem for the coefficients of the partition generating function. See, for example, the saddle-point analysis in Andrews (1976, Chapter 6).

² Partitions are conventionally written in non-increasing order, though the order of parts does not matter.

³ The conjugation symmetry of partitions, via Ferrers diagrams, is a standard result in partition theory. See Andrews (1976, Chapter 1) for a detailed exposition.

⁴ This is the classical generating function for partitions, due to Euler. The identity follows from the expansion , where each term represents a choice of how many parts of size appear.

⁵ The constructive method—building partitions by leading digit—is the top-down interpretation of the DP recurrence . The leading digit corresponds to using at least one part of size , with the remainder .

⁶ This asymptotic is derived from the Erdős–Lehner theorem, which states that the largest part of a random partition of concentrates near . The mode of follows the same asymptotic location.

⁷ Erdős and Lehner (1941). A modern treatment is given in Fristedt (1993), who developed the general framework for random partitions.

⁸ See Andrews (1976, Chapter 6) for the saddle-point method applied to the generating function , which yields the local normality of the coefficients.

⁹ Pearson's mechanical models, in which he generated distributions by physical drawings and discrete approximations, represent an early example of constructive statistics. See Pearson (1895) for his original presentation.

References

  1. Andrews, G.E. (1976). The Theory of Partitions. Cambridge University Press.
  2. Erdős, P. and Lehner, J. (1941). The distribution of the number of summands in the partitions of a positive integer. Duke Mathematical Journal, 8(2), 335–345.
  3. Fristedt, B. (1993). The structure of random partitions of large integers. Transactions of the American Mathematical Society, 337(2), 703–735.
  4. Pearson, K. (1895). Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material. Philosophical Transactions of the Royal Society A, 186, 343–414.

 

 


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