All Questions

Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...

Define a $k$-permutation of $\{1,\ldots, n\}$ to be a word $\tau_1 \ldots \tau_k$ such that $\{\tau_1,\ldots,\tau_k\}$ is a $k$-subset of $\{1,\ldots, n\}$. Thus an $n$-permutation of $\{1,\ldots, n\}$...

Conjecture: Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$. An enumeration $ E \cup -E = \{f_1, \ldots, ...

Consider a totally ordered alphabet $A$ of $n$ letters. Let $W$ be the set of all words over $A$ which have no two letters equal. Then, for example, we can define the Knuth equivalence on $W$ as the ...