arXiv subjects

Jaideep Sai Padhi

Publications and source records attributed to Jaideep Sai Padhi.

2 recordsLinked to original sources

arXiv2026

Sign patterns of real powers of infinite products: resolution of four conjectures of Schlosser and Zhou

For an infinite product P(q) = prod_{m>=1} (1-q^m)^{eps(m)} and real delta, write P(q)^delta = sum_{n>=0} c_delta(n) q^n. Schlosser and Zhou conjectured precise sign patterns for these coefficients, for several products and ranges of delta. We resolve four of their conjectures completely, in each case extending partial results already in the literature: those for the Goellnitz-Gordon product Q_8 = (q,q^7;q^8)_inf/(q^3,q^5;q^8)_inf (Conjecture 21), for Q_12 = (q,q^11;q^12)_inf/(q^5,q^7;q^12)_inf (Conjecture 24), and for the Borwein products G_7 and G_11 (Conjectures 20 and 23). Conjecture 23 is true. Conjecture 21 fails exactly on [beta, 8/3), where beta is approximately 2.66448; the threshold 8/3 is sharp, and the first counterexamples occur near n = 7.6 x 10^5. Conjecture 24 fails exactly on (delta_1, 0), where delta_1 = -0.64411... is a root of the 22nd coefficient polynomial. Conjecture 20 fails exactly on (delta_c, 5), where delta_c = 4.8735075853867342634... is a root of the 897th coefficient polynomial. All other stated ranges are proved. Parts of Conjecture 21 were settled independently by He and Li; what is new here is the range near delta = -1, where their threshold diverges. Each failure has the same mechanism: a leading circle-method amplitude vanishes, and a secondary term with the wrong sign overtakes it. The proofs are self-contained apart from classical facts. The cusp analysis is exact, via finite orbits of Weil representations; the Hardy-Ramanujan-Rademacher expansion is made fully explicit; and certified computations in exact and ball arithmetic handle the finite ranges and the degenerate regimes.

math.CO↗

arXiv2026

Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics, with an Account of the Human-Machine Methodology Employed

We report solutions to five challenge problems posed by Doron Zeilberger and his collaborators, together with substantial partial progress on two further problems, and we describe the method by which they were obtained. The solved problems are: the Second Computational Chomp Challenge of Ekhad and Zeilberger, for which we exhibit a bar with three winning opening moves; the third challenge of Spahn and Zeilberger, asking whether the restricted permutation counts a_{r,s} and b_{r,s} are holonomic for all r,s>1, answered affirmatively; the First Rigorous Solid Standard Young Tableaux Challenge, for which we prove the conjectured second-order recurrence; the five-dimensional Geode Challenge of Amdeberhan, Kauers and Zeilberger; and Conjectures 2a and 2b of Kauers and Zeilberger, which we obtain from a local limit theorem for excursions of Markov-modulated random walks in cones. Several results of independent interest arise along the way: a staircase theorem constraining the winning opening moves of any Chomp bar, together with a parity theorem for square bars; an explicit algebraic generating function for reverse-Kreweras diagonal walks and a closed form for their diagonal-endpoint counts; a one-dimensional integral representation for diagonal Geode coefficients; and the identity that each Kauers-Zeilberger constant is a universal factor times the square of the apex value of a discrete cone-harmonic function. All of the work reported here was carried out in collaboration with a large language model. The paper sets out the division of labour, records the verification protocol this mode of work required, and documents the failures, which we regard as an essential part of the report.

math.CO↗