On Scores, Losing Scores and Total Scores in Hypertournaments

A $k$-hypertournament is a complete $k$-hypergraph with each $k$-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a $k$-hypertournament, the score $s_{i}$ (losing score $r_{i}$) of a vertex $v_{i}$ is the number of arcs containing $v_{i}$ in which $v_{i}$ is not the last element (in which $v_{i}$ is the last element). The total score of $v_{i}$ is defined as $t_{i}=s_{i}-r_{i}$. In this paper we obtain stronger inequalities for the quantities $\sum_{i\in I}r_{i}$, $\sum_{i\in I}s_{i}$ and $\sum_{i\in I}t_{i}$, where $I\subseteq \{ 1,2,\ldots,n\}$. Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong $k$-hypertournaments

Sumwords: Record Scores and Anagrams

I first present a list of just over a hundred sumgrams (words with equal numerical value) of the longest words I was able to conquer. Note that one pair of reasonably common words could be construed as opposites: EXTRAORDINARINESS and STRAIGHTFORWARDNESS. Note also that all pairs of words with endings -IST and -ISMS are sumgrams, as the values of M and S sum to the value of T. I have included the longest such specimens in my list