arXiv subjects
Publications and source records attributed to Vsevolod Evtushevsky.
For the jump graph on the subgraph of the Young--Fibonacci graph formed by words with at most $K$ twos, formulas for the number of paths between vertices are obtained and the Martin boundary of the path space is described.
For the $r$-differential version of the Young--Fibonacci graph, formulas for the number of paths between vertices are obtained; the Martin boundary of the path space is described; and the ergodicity of the corresponding measures is proved.
On the set $\mathbb{YF}_n$ of words in $1$ and $2$ with digit sum $n$, we consider the probability distribution $μ_P$, corresponding to the Plancherel measure on paths in the Young--Fibonacci graph. We study the asymptotics as $n$ grows of the entropy of the distribution $μ_P$ and of the minimum number of words on which $μ_P$ is almost entirely concentrated.
For a poset $(P,\leqslant)$ we consider the first-order theory, that is defined by set $P$ and relation $\leqslant$. The problem of undecidability of combinatorial theories attracts significant attention. Recently A. Wires proved the undecidability of the elementary theory of Young lattice and also established the maximal definability property of this theory. The purpose of this article is to obtain the same results for another graded lattice, which has much in common with Young lattice: Young--Fibonacci lattice. As Wires does for Young lattice, for the proof of undecidability we define Arithmetic into this theory.
Among central measures on the path space of the Young--Fibonacci lattice the so-called Plancherel measure has a special role. Its ergodicity was proved by Kerov and Gnedin. The goal of this cycle of two articles is to prove that remaining measures from the Martin boundary of this graph (which were described by Kerov and Goodman) are also ergodic. The measures are parametrized with an infinite word of digits 1 and 2 and the parameter $β\in(0,1]$ (the case $β=0$ corresponds to the Plancherel measure). In this article we prove the statements which correspond to the case $β=1$.
Among central measures on the path space of the Young--Fibonacci lattice the so-called Plancherel measure has a special role. Its ergodicity was proved by Kerov and Gnedin. The goal of this cycle of two articles is to prove that remaining measures from the Martin boundary of this graph (which were described by Kerov and Goodman) are also ergodic. The measures are parametrized with an infinite word of digits 1 and 2 and the parameter $β\in(0,1]$ (the case $β=0$ corresponds to the Plancherel measure). In this article we finish the proof of their ergodicity using the statements proved in the first paper as a "black box".
The Young--Fibonacci graph is the Hasse diagram of one of the two (along with the Young lattice) 1-differential graded modular lattices. This explains the interest to path enumeration problems in this graph. We obtain a formula for the number of paths between two vertices of the Young--Fibonacci graph which is polynomial with respect to the minimum of their ranks.