Combinatorics and Graph Theory

2610 Submissions

[2] ai.viXra.org:2610.0060 [pdf] submitted on 2026-10-26 23:59:56

The Internal Space and Boundary Growth of the Spectre and Tsiokos’s Chair44 Tilings

Authors: Brent Hartshorn, Tristan Romanov
Comments: 30 Pages.

The planar Spectre and Tsiokos’s three-dimensional Chair44 are aperiodic monotiles whose tilings are hierarchical; they are compared here by exact computation. The Spectre inflates by the unit λ2 = √15 + 4 of ℤ[√15]. Its supertile perimeter is 2F3k+5+4 at every order k, proved by a finite exact certificate, so it grows by the unit 2 + √5 of ℚ(√5), faster than the linear inflation. The boundary of the limiting supertile is a fractal of box dimension log(2 + √5)/ log λ, and the tail of the common-ancestor levels of neighbouring tiles decays exactly by ν/λ2 . The inflation acts on the vertex module ℤ4 with two expanding and two contracting directions, and the internal plane is given exactly; the window is the attractor of contracting similarities, with boundary dimension log(5+2√6)/log(4+√15). The Chair44 inflates by the integer 2 on ℤ3 , with no Euclidean internal space (its internal space is 2-adic) and no twist; its centre set, that of the classical three-dimensional chair, has an explicitly computed pure-point diffraction measure. The main result is that every lattice-registered tiling obeying the 44 contact rules lies in the substitution hull, and that its symmetry group is conjugate to one of 1, C2 (a half-turn about a face diagonal), C4 and D4. Both statements use Kamenetsky’s unique-hierarchy criterion, whose finite certificate is checked here independently, and they transfer to every tiling by the Chair44 solid through Tsiokos’s registrationtheorem, which has been audited but not refereed. Orientation sums and contact matrices separate the two tilings. Four finite-patch computations then test whether the Spectre’s substitution hierarchy can serve as a clock: positional clustering shows no arrow, the radius needed to recognize an ancestor grows about threefold per level without making one, the look-back law is shared with a quadtree, and the parity flip of the construction shows in the outlines of supertiles and not in the tiles.
Category: Combinatorics and Graph Theory

[1] ai.viXra.org:2610.0002 [pdf] submitted on 2026-10-01 15:48:26

Asymptotics of the Variance in the Collector’s Problem

Authors: Bolok Li
Comments: 22 Pages.

In the collector's problem with group drawings, one run draws a uniformly random q-subset of an n-element set, all q coupons being kept, and T denotes the number of runs needed to complete the set. The distribution of T and all of its factorial moments are due to Stadje (1990); we do not claim them, and we say explicitly where our formulas reproduce his -- in particular his relation (2.15) at p = 2 already yields the second moment, so item (i) below is a consolidation rather than a new formula. The asymptotics of E[T] as n -> infinity with q fixed are due to Berend and Sher (2026) and, in a sharper form, to Doumas and Spektor (2026); we do not claim those either, and Section 8 downgrades our own expectation expansion to an independent rediscovery accordingly. What we add is fourfold, of which the first is a consolidation rather than a new formula.(i) An exact closed form for the second moment, hence for the variance, obtained by summing the inclusion-exclusion tail in closed form, together with a uniform one-line derivation of every higher moment. Having now read Stadje (1990) in full we record that his general factorial-moment relation (his eq. (2.15)), specialised to A = S and p = 2, is equivalent to our second-moment formula; item (i) is therefore an explicit specialisation of Stadje's result and is not claimed as new.(ii) Two subtraction-free recursions -- a backward recursion for E[T] in O(nq) time and O(q) memory, and a forward recursion for the whole distribution -- which keep full double-precision accuracy up to n = 1000, whereas the inclusion-exclusion closed form loses all significant digits from n about 40 onwards: at (n,q) = (60,5) its relative error is 4.9 x 10^{-4} and at (80,5) it returns 1.99 x 10^{6} for the true value 77.836.(iii) A quantitative explanation: the relative error of the closed form is predicted by eps*kappa, where eps is the unit roundoff and kappa = sum_i |t_i| / |sum_i t_i| is the condition number of the alternating sum; measured and predicted errors agree to within two orders of magnitude over twenty-six decades.(iv) Asymptotics of the variance, with a complete proof. Writing A = H_n - H_{n-q}, Sigma_2 = sum_{j 0,hence the n log n formVar[T] = (pi^2/6)(n/q)^2 - (n/q^2)(log n + gamma) - ( ((pi^2/6)(q-1) + 1)/q^2 ) n + ((q-1)/(2q^2)) log n + c_0(q) + O(log n / n).For q = 1 everything degenerates exactly to the classical Var[T] = n^2 H_n^{(2)} - n H_n. Doumas and Spektor obtain for the variance only the leading term (pi^2/6)(n/q)^2 (1 + o(1)), so everything beyond it — the n log n term, the n-coefficient, the constant, the closed form and the exponential rate -- is new for q >= 2 as far as we have been able to determine; at q = 1 the expansion collapses to the known result of Doumas and Papanicolaou (2012). We also record the exact support [ceil(n/q), infinity) and point out that the widely quoted formula of Sharif-Hassibi and Xu-Tang belongs to a different model (draw d, keep one), which is why it gives different numbers.
Category: Combinatorics and Graph Theory