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Two binomial inequalities prove the unimodal, asymmetric shape of heady score counts for binary strings.

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2026-07-10 16:59 UTC pith:VZHGEBD2

load-bearing objection Solid completion of two binomial-expectation inequalities that finish the unimodality proof for the author’s heady-s counts; the only real soft spot is a finite-n numerical bridge for three small u values. the 2 major comments →

arxiv 2607.07837 v1 pith:VZHGEBD2 submitted 2026-07-08 math.CO

On Two Combinatorial Inequalities That Explain the Blimpy Shape of Heady-s and Taily-s Bit Strings

Bruce Levin This is my paper
classification math.CO MSC 05A2060C05
keywords combinatorial inequalitiesgenerating functionsbinomial coefficientsunimodalityexpectation inequalitiesPascal's trianglebit stringssingularity analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves two combinatorial inequalities that finish the explanation of the dirigible-like graph of the number of fixed-length bit strings carrying a given net score under the Alice–Bob scoring rule (Alice scores on consecutive heads, Bob on head-then-tail). Generating functions and singularity analysis establish a lower bound on an expectation formed from products of binomial coefficients, and a comparison of weighted averages of binomial coefficients along adjacent oblique rays in Pascal’s triangle. These facts imply that the count H_s(n) is unimodal, peaking at score zero for heady strings, and that the positive-score side is elongated relative to the negative-score side. The result therefore accounts for both the central peak and the left–right asymmetry visible for n = 100.

Core claim

For every admissible n and s the heady count H_s(n) satisfies H_s(n) ≥ H_{s+1}(n) when s ≥ 0 and H_s(n) ≥ H_{s-1}(n) when s ≤ 0. The proof reduces the claim to two inequalities: E[K] ≥ (n − s − 1)/4 under the double-factor measure proportional to binom(s+k,2k) binom(n−k,2k), and a ray-average comparison of binomial coefficients for non-positive scores with offsets 0, 1 and 2.

What carries the argument

Ordinary generating functions for the sequences of binomial products, analysed by the Flajolet–Sedgewick singularity-transfer theorems that extract the asymptotic growth of the coefficients and thereby prove the expectation and ray inequalities for large n.

Load-bearing premise

For non-positive scores with offsets 0, 1 and 2 the analytic argument only covers sufficiently large n, so the complete statement rests on direct numerical verification of the finite sums up to several hundred.

What would settle it

Evaluate the double-factor expectation E[K] against (n−s−1)/4 for many pairs (n,s ≥ 0) and check whether the inequality ever fails; or compute successive ratios H_s(n)/H_{s+1}(n) for moderate n across the full range of s and look for a unimodality violation.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Unimodality of H_s(n) holds for all n ≥ 2 and every admissible score s, completing the earlier partial proof.
  • The same unimodality transfers at once to the taily counts, which therefore peak at score −1.
  • The positive-side elongation is forced by the truncation of the summation index that appears in the explicit formula for H_s(n).
  • The ray comparison supplies a new combinatorial fact about weighted averages of binomial coefficients along parallel oblique lines in Pascal’s triangle.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Analogous ray inequalities may hold for other binomial weights and could explain unimodality or skewness in related pattern-counting distributions.
  • The same generating-function expansions can be pushed one order further to obtain concentration or variance bounds on the score random variable.
  • The numerical bridge up to n = 500 strongly suggests the ray inequality is true for every n, inviting a fully closed-form proof that removes the case split on the score offset.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves two combinatorial inequalities that complete the proof of unimodality of the heady-s counts H_s(n) (Theorem 1.1), thereby explaining the unimodal and asymmetric “blimpy” shape of the score distribution for Alice–Bob bit-string scores. For non-negative scores the key result is Theorem 2.1: under the double-factor measure proportional to C(s+k,2k)C(n-k,2k) one has E[K]≥(n-s-1)/4 (strict except for two small cases). For non-positive scores the paper establishes the ray-average inequality (1.18)/Theorem 3.1 for u=0,1,2 and n large, which a fortiori yields the necessary-and-sufficient condition (1.12); the remaining cases u>2 and small n are imported from the author’s earlier work L25. Both inequalities are obtained from ordinary generating functions (Appendices A–B) via singularity analysis / transfer theorems, supplemented by a Chebyshev covariance argument for s>0 and direct numerical verification of the finite binomial sums C_{n,u}, D_{n,u} for intermediate n.

Significance. The inequalities are of independent combinatorial interest (expectation bounds for products of binomial coefficients and comparisons of weighted averages along parallel oblique rays in Pascal’s triangle). They close the remaining analytic gaps left by L25 and thereby give a complete generating-function proof of the unimodality statement that underlies the graphical shape of the Alice–Bob score distribution. The derivations are standard, carefully written, and largely self-contained; the positive-score side is fully rigorous for all n≥3. The work therefore supplies a clean, reusable analytic toolkit for a concrete family of discrete distributions arising from a popular probability puzzle.

major comments (2)
  1. Theorem 3.1 and the subsequent claim that Theorem 1.1 holds for all n under (1.10) rest on a computational bridge: singularity analysis only guarantees (3.8) for “sufficiently large n”, after which the paper invokes direct evaluation of the finite sums C_{n,u} and D_{n,u} up to n=500 (with claimed thresholds n≥8,13,23 for u=0,1,2). While the numerical checks appear thorough and the asymptotic regime is entered well before n=500, a fully analytic statement of Theorem 1.1 would require either an explicit closed-form verification for the finite intermediate range or a sharper remainder estimate that covers all n≥7. The present hybrid argument is load-bearing for completeness under condition (1.10).
  2. The cases u>2 and the elementary small-n arguments are imported wholesale from L25 (Lemmas 6.1–6.8 and Remark 3.1). For a self-contained journal article it would be preferable either to reproduce the short elementary verifications or to state precisely which statements of L25 are being used as black boxes, so that a reader can verify the logical chain without consulting the earlier paper.
minor comments (4)
  1. Notation for the length parameter is overloaded: after the re-indexing at the start of §2, n stands for the former n_s=n-s-1, while in §3 it reverts to the original bit-string length. A short clarifying sentence or a distinct symbol would reduce the risk of confusion.
  2. Figure 1 is reproduced only as a large table of exact integers; a conventional log-scale plot (or a pointer to the earlier L25 figure) would make the “blimpy” shape immediately visible to the reader.
  3. The APL programs mentioned in Remark 2.1 are not deposited; a short supplementary notebook or a public repository link would improve reproducibility of the numerical thresholds.
  4. A few typographical slips appear (e.g., “peri-central”, “not-quite-complete”, occasional missing spaces around operators). A careful copy-edit pass would clean them up.

Circularity Check

0 steps flagged

No significant circularity: generating-function proofs of the two inequalities are independent of the unimodality target; only minor self-citation of combinatorial setup from L25.

full rationale

The paper’s central results (Theorem 2.1 lower-bounding E[K] under the double-factor measure, and Theorem 3.1 establishing the ray-average inequality (1.18) for u=0,1,2) are derived from ordinary generating functions (Lemmas A.1–A.3, B.1–B.4), singularity-analysis transfer theorems (A.2)–(A.4) and (B.2), and elementary a-fortiori/Jensen arguments. These steps do not presuppose unimodality of H_s(n). Self-citations to L24/L25 supply only the scoring definition, the explicit sum (1.1), and the already-derived necessary-and-sufficient conditions (Lemmas 1.1–1.2); they are not used to force the inequalities themselves. The finite-n numerical verification of C_{n,u} gtr D_{n,u} for intermediate n is computational confirmation, not a fitted parameter renamed as a prediction. No self-definitional loop, uniqueness import, or ansatz smuggling appears. Score 1 reflects only the ordinary (non-load-bearing) self-citation of prior combinatorial setup.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 1 invented entities

The work is pure enumerative combinatorics. It rests on standard binomial identities, ordinary generating functions, Flajolet–Sedgewick singularity analysis, Chebyshev’s covariance inequality, Jensen’s inequality, and Pascal’s identity. Domain setup (heady/taily strings, score S, formulas for H_s(n)) is imported from the author’s L24/L25. No continuous free parameters are fitted; the only non-analytic residue is finite-n numerical verification for three small values of u.

axioms (7)
  • standard math Flajolet–Sedgewick transfer theorems for coefficients of (1−x/β)^−α F(x) with F analytic at β (Theorem VI.1 and basic expansion (A.1)–(A.4)).
    Used throughout Appendices A and B to extract the n^{−1/2} and n^{−3/2} asymptotics of A_n, B_n, C_{n,u}, D_{n,u}.
  • standard math Chebyshev’s covariance inequality: Cov(K,g(K)) > 0 when g is increasing and non-constant on the support.
    Invoked in the proof of Theorem 2.1 to show E_{n,s}[K] is strictly decreasing in s.
  • standard math Jensen’s inequality for the convex function f appearing after the change of measure to the double-factor law (1.7).
    Converts the mean lower bound (1.8) into non-negativity of E[f(K)], which is equivalent to unimodality for s ≥ 0.
  • standard math Closed-form and recursive generating functions for central and near-central binomial sequences (Lemmas A.1–A.3, B.1–B.4).
    Derived in the appendices from binomial series and linear recurrences; standard but re-derived for completeness.
  • domain assumption Formulas (1.1) and (1.9) for H_s(n) and the necessary-and-sufficient conditions of Lemmas 1.1–1.2 (from L25).
    The entire reduction of unimodality to the two inequalities rests on these combinatorial identities and equivalences established in the prior paper.
  • domain assumption For u > 2 the sufficient mean bound (1.17) already holds for all n ≥ 7 (L25, Lemmas 6.1–6.8).
    Allows the present paper to treat only u = 0,1,2 with generating functions.
  • ad hoc to paper Numerical evaluation of the finite binomial sums C_{n,u}, D_{n,u} correctly decides the inequalities for all n from the tabulated thresholds up to 500.
    Bridges the gap between the asymptotic regime and the smallest n under condition (1.10); not a formal proof for every intermediate n.
invented entities (1)
  • Heady-s / taily-s bit strings and the Alice–Bob net score S independent evidence
    purpose: Define the combinatorial objects whose count histogram is being explained.
    Introduced in the author’s prior L24/L25 work; not new here, but they are the domain objects the inequalities serve.

pith-pipeline@v1.1.0-grok45 · 44942 in / 3641 out tokens · 41273 ms · 2026-07-10T16:59:45.221241+00:00 · methodology

0 comments
read the original abstract

We prove two inequalities introduced in our prior study of the graphical shape of the number of bit strings with a given score under an interesting scoring system. Generating functions are used to establish the inequalities, which in turn imply two of the salient graphical features, uni-modality near the zero score and shape asymmetry for positive versus negative scores. One inequality provides a lower bound on the expected value of a discrete random variable with probabilities proportional to a product of two binomial coefficients. The other inequality states that the expected value with respect to near central binomial coefficients of other binomial coefficients lying on an oblique ray in Pascal's triangle exceeds the expected value along an adjacent parallel ray to its left.

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages · 3 internal anchors

  1. [1]

    On cases where Litt's game is fair

    Basdevant, A.-L., Hénard, O., Maurel-Segala, E. and Singh, A. (2024). On cases where Litt’s game is fair. arXiv:2406.20049, https://doi.org/10.48550/arXiv.2406.20049

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2406.20049 2024

  2. [2]

    How to Answer Questions of the Type: If you toss a coin n times, how likely is HH to show up more than HT?

    Ekhad, S.B. and Zeilberger, D. (2024). How to answer questions of the type: If you toss a coin n times, how likely is HH to show up more than HT? arXiv:2405.13561, https://doi.org/10.48550/arXiv.2405.13561

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2405.13561 2024

  3. [3]

    and Sedgewick, R

    Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge: Cambridge University Press

    work page 2009

  4. [4]

    Janson, S., Nika, M., and Segert, S. (2025). The generalized Alice HH and Bob HT problem. arXiv:2503.19035, https://doi.org/10.48550/arXiv.2503.19035

    work page doi:10.48550/arxiv.2503.19035 2025

  5. [5]

    Levin, B. (2024). Note on a coin-tossing problem posed by Daniel Litt. arXiv:2409.13087, https://doi.org/10.48550/arXiv.2409.13087

    work page doi:10.48550/arxiv.2409.13087 2024

  6. [6]

    Levin, B. (2025). The blimpy shape of heady-s and taily-s bit strings. Sequential Analysis, advance online publication, https://doi.org/10.1080/07474946.2025.2529212. Print journal published 2006, 45(1):1-53

    work page doi:10.1080/07474946.2025.2529212 2025

  7. [7]

    Litt, D. (2024). X-post, March 16, https://x.com/littmath/status/1769044719034647001

    work page arXiv 2024

  8. [8]

    Segert, S. (2024). A proof that HT is more likely to outnumber HH than vice sersa in a sequence of n coin flips. arXiv:2405.16660, https://doi.org/10.48550/arXiv.2405.16660. Figure 1. Number of heady-s (left) and taily-s (right) 100-bit strings with score sxS =)( )100( (center)               ...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2405.16660 2024