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Register a new userLazy tournaments and multidegrees of a projective embedding of $overline{M}_{0,n}$
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We provide a new geometric interpretation of the multidegrees of the (iterated) Kapranov embedding $Phi_n:overline{M}_{0,n+3}hookrightarrow mathbb{P}^1times mathbb{P}^2times cdots times mathbb{P}^n$, where $overline{M}_{0,n+3}$ is the moduli space of stable genus $0$ curves with $n+3$ marked points. We enumerate the multidegrees by disjoint sets of boundary points of $overline{M}_{0,n+3}$ via a combinatorial algorithm on trivalent trees that we call a lazy tournament. These sets are compatible with the forgetting maps used to derive the recursion for the multidegrees proven in 2020 by Gillespie, Cavalieri, and Monin. The lazy tournament points are easily seen to total $(2n-1)!!=(2n-1)cdot (2n-3) cdots 5 cdot 3 cdot 1$, giving a natural proof of the fact that the total degree of $Phi_n$ is the odd double factorial. This fact was first proven using an insertion algorithm on certain parking functions, and we additionally give a bijection to those parking functions.
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For any two squares A and B of an m x n checkerboard, we determine whether it is possible to move a checker through a route that starts at A, ends at B, and visits each square of the board exactly once. Each step of the route moves to an adjacent square, either to the east or to the north, and may step off the edge of the board in a manner corresponding to the usual construction of a projective plane by applying a twist when gluing opposite sides of a rectangle. This generalizes work of M.H.Forbush et al. for the special case where m = n.
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We describe computer searches that prove the graph reconstruction conjecture for graphs with up to 13 vertices and some limited classes on larger sizes. We also investigate the reconstructability of tournaments up to 13 vertices and posets up to 13 points. In all cases, our proofs also apply to the set reconstruction problem that uses the isomorph-reduced deck.
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We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(ell)$ be the limit of the ratio of the maximum number of cycles of length $ell$ in an $n$-vertex tournament and the expected number of cycles of length $ell$ in the random $n$-vertex tournament, when $n$ tends to infinity. It is well-known that $c(3)=1$ and $c(4)=4/3$. We show that $c(ell)=1$ if and only if $ell$ is not divisible by four, which settles a conjecture of Bartley and Day. If $ell$ is divisible by four, we show that $1+2cdotleft(2/piright)^{ell}le c(ell)le 1+left(2/pi+o(1)right)^{ell}$ and determine the value $c(ell)$ exactly for $ell = 8$. We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length $ell$ when $ell$ is not divisible by four or $ellin{4,8}$.
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