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Threshold counts in overpartitions

Abstract

A finite identity relates the full threshold statistic on native overpartitions to total and nonoverlined part counts.

Nat denotes the nonnegative integers and Int the integers. An ordinary partition is a multiset of positive integers with sum n. Its support contains its distinct part sizes. Each subset S of that support selects the sizes whose first occurrence is overlined, independently, so every legal marking pattern occurs exactly once. The functions parts and marks below select these two components; indicator is one when its proposition holds and zero otherwise. Every displayed sum and cardinal is finite.

Definition 1.1 (The native marked carrier).

Formalization. D5/S3/Combinatorics/Partitions/OverpartitionThreshold.Overpartition (✓ std3).

Citation. Subhash Chand Bhoria, Pramod Eyyunni, Subhrangsu Santra (2026). On the number of missing integers in partitions. DOI: 10.48550/arXiv.2604.12557. URL: https://arxiv.org/abs/2604.12557v1.

Commentary.

Membership of a pair is shorthand for the dependent pair of an ordinary partition and a subset of its support. There are no additional marking constraints.

Definition 1.2 (Multiplicity includes the mark).

Formalization. D5/S3/Combinatorics/Partitions/OverpartitionThreshold.multiplicity (✓ std3).

Citation. Subhash Chand Bhoria, Pramod Eyyunni, Subhrangsu Santra (2026). On the number of missing integers in partitions. DOI: 10.48550/arXiv.2604.12557. URL: https://arxiv.org/abs/2604.12557v1.

Commentary.

count counts all occurrences in the underlying multiset. Overlining changes none of these counts.

Definition 1.3 (Largest part).

Formalization. D5/S3/Combinatorics/Partitions/OverpartitionThreshold.largest (✓ std3).

Citation. Subhash Chand Bhoria, Pramod Eyyunni, Subhrangsu Santra (2026). On the number of missing integers in partitions. DOI: 10.48550/arXiv.2604.12557. URL: https://arxiv.org/abs/2604.12557v1.

Commentary.

sup is the maximum of the finite support, with value zero on the empty support.

Definition 1.4 (The full threshold statistic).

Formalization. D5/S3/Combinatorics/Partitions/OverpartitionThreshold.U (✓ std3).

Citation. Subhash Chand Bhoria, Pramod Eyyunni, Subhrangsu Santra (2026). On the number of missing integers in partitions. DOI: 10.48550/arXiv.2604.12557. URL: https://arxiv.org/abs/2604.12557v1.

Commentary.

The interval is closed and begins at one. Missing sizes have multiplicity zero, so they are counted whenever k is positive. For n equal to zero the interval is empty.

Definition 1.5 (Total parts).

Formalization. D5/S3/Combinatorics/Partitions/OverpartitionThreshold.T (✓ std3).

Citation. Hartosh Singh Bal, Gaurav Bhatnagar (2021). Stanley–Elder–Fine theorems for colored partitions. DOI: 10.48550/arXiv.2102.03486. URL: https://arxiv.org/abs/2102.03486v2.

Commentary.

card(parts(o)) is the number of all occurrences, including marked ones; equivalently it is the sum of their multiplicities over positive sizes.

Definition 1.6 (Nonoverlined occurrences).

Formalization. D5/S3/Combinatorics/Partitions/OverpartitionThreshold.N (✓ std3).

Citation. Hartosh Singh Bal, Gaurav Bhatnagar (2021). Stanley–Elder–Fine theorems for colored partitions. DOI: 10.48550/arXiv.2102.03486. URL: https://arxiv.org/abs/2102.03486v2.

Commentary.

A selected size contributes exactly one overlined occurrence. Subtracting its indicator leaves the actual nonoverlined occurrences.

Theorem 1.7 (The uniform finite identity).

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Partitions/OverpartitionThreshold.result (✓ std3). ∎

Resolves. Problems/bhoria-eyyunni-santra-2026-overpartition-threshold-identity (proved) by D5/S3/Combinatorics/Partitions/OverpartitionThreshold.result.

Source. Repository-derived.

Acknowledgement. Subhash Chand Bhoria, Pramod Eyyunni, Subhrangsu Santra (2026). On the number of missing integers in partitions. DOI: 10.48550/arXiv.2604.12557. URL: https://arxiv.org/abs/2604.12557v1.

Acknowledgement. Joselyne Aniceto, Cristina Ballantine (2026). Identities involving the number of missing integers in partitions - combinatorial proofs. DOI: 10.48550/arXiv.2608.00278. URL: https://arxiv.org/abs/2608.00278v1.

Acknowledgement. Hartosh Singh Bal, Gaurav Bhatnagar (2021). Stanley–Elder–Fine theorems for colored partitions. DOI: 10.48550/arXiv.2102.03486. URL: https://arxiv.org/abs/2102.03486v2.

Commentary.

For every positive k and every nonnegative n the equality is in Int. The alternating sum is empty for k greater than n. Ordinary Ferrers conjugation preserves the number of distinct sizes, so its reindexing preserves the number of marking subsets and exchanges the largest-part total with the total number of parts. Removing k copies of one positive size gives the threshold recurrence. Toggling the optional mark and removing a nonoverlined occurrence give the frequency recurrence; finite alternating cancellation and the pointwise complementary count yield the displayed equality. The overpartition decomposition, nonoverlined frequency formula and threshold-one and threshold-three ingredients have earlier literature precedents. The cited Question 3 asks for a concise analogue, and the later Conclusion retains that question; neither source asserts this particular formula.

References