Integer sequences and combinatorics

Results in Integer sequences and combinatorics

• Order-polynomial positivity for width-two posets with a chain of at most five elements

  Partially resolvedFerroni–Morales–Panova conjecture on positivity of order polynomials of width-two posets

  Mathematics · Integer sequences and combinatorics · standalone Lean package

  If a finite poset P is the union of two chains, one with at most five elements and the other arbitrary, then |P|! Ω_P(t) has nonnegative coefficients, and [t]Ω_P(t) ≥ (a − 1)!(b − 1)! m/(a + b)!.

  Manuscript: Order-polynomial positivity for width-two posets with a chain of at most five elements · Lean proof: Order-polynomial positivity for width-two posets with a chain of at most five elements · Verification: Order-polynomial positivity for width-two posets with a chain of at most five elements

• Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions

  DisprovedLetouzey–Li–Steiner late-iterate ordering conjecture for generalized Hofstadter functions

  Mathematics · Integer sequences and combinatorics · shared Lean library

  For every k ≥ 4 some input n has Fₖ^(2k+1)(n) > Fₖ₊₁^(2k+2)(n), contradicting the conjectured order Fₖ^j ≤ Fₖ₊₁^(j+1) for j > 2k; an explicit witness is F₄⁹(132) = 8 > 7 = F₅¹⁰(132).

  Manuscript: Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions · Lean proof: Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions · Verification: Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions

• Sharp root log-concavity of alternating Motzkin sums

  ProvedZhao's root log-concavity conjecture for alternating sums of Motzkin numbers

  Mathematics · Integer sequences and combinatorics · shared Lean library

  With Tₙ the alternating sum of the Motzkin numbers M₀,…,Mₙ, the sequence Tₙ^(1/n) is strictly log-concave from n = 5 and the inequality reverses at center 5, so 5 is optimal; this contains Zhao's tail from n = 9.

  Manuscript: Sharp root log-concavity of alternating Motzkin sums · Lean proof: Sharp root log-concavity of alternating Motzkin sums · Verification: Sharp root log-concavity of alternating Motzkin sums

• Sharp log-concavity for alternating Motzkin sums

  New theoremBounded-forcing log-concavity for Motzkin-type recurrences and alternating Motzkin sums

  Mathematics · Integer sequences · standalone Lean package

  Real aₙ with (m+2)aₘ = m(2aₘ₋₁+3aₘ₋₂)+εₘ, |εₘ| ≤ 2 for m ≥ 14, a₁₂ > 1800 and L₁₂ < a₁₃/a₁₂ < U₁₂ satisfy aₙ > 8(n+3)² and a ratio log-concavity margin 800aₙ⁴/(n+3)³ for all n ≥ 12.

  Manuscript: Sharp log-concavity for alternating Motzkin sums · Lean proof: Sharp log-concavity for alternating Motzkin sums · Verification: Sharp log-concavity for alternating Motzkin sums

• A positive constant-recursive sequence with nonquasipolynomial power ranks

  CounterexamplePower ranks of constant-recursive sequences: Rowland–Sistos Barrón finiteness conjecture

  Mathematics · Integer sequences · standalone Lean package

  s(n) = 1 + 2^n + 3^n(1 − 2^n)² + n9^n(1 + 2^n)³ has rank 13 and power ranks P(M) − 2Q(M), not eventually quasipolynomial; a companion family gives infinitely many power-rank sequences at each rank r ≥ 3.

  Manuscript: A positive constant-recursive sequence with nonquasipolynomial power ranks · Lean proof: A positive constant-recursive sequence with nonquasipolynomial power ranks · Verification: A positive constant-recursive sequence with nonquasipolynomial power ranks