Color sum

The color sum of a two-colored partition $p\in\Pscr^{\circ\bullet}$ is the integer-valued measure $\sigma_p:\mathfrak{P}(P_p)\to\Z$ on its set of points $P_p$ whose density assigns to any point $i\in P_p$ the value $1$ if $i$ is of normalized color $\circ$ in $p$ and $-1$ if $i$ is of normalized color $\bullet$ in $p$ (see [MaWe19], Section 3.3).

Per definition, if $i$ is a lower point, then the normalized and the ordinary color of $i$ in $p$ coincide, and the two are opposites if $i$ is an upper point.

The color sum $\Sigma(p)\colon\hspace{-0.66em}=\sigma_p(P_p)$ of the set of all points of $p$ is called the total color sum of $p$ (see [TaWe18], Definition 2.4).

A set $S\subseteq P_p$ of points of $p$ is said to be neutral if its color sum vanishes, $\sigma_p(S)=0$ (see [MaWe19], Section 3.3).

References

[MaWe19] Mang, Alexander and Weber, Moritz, 2019. Categories of two-colored pair partitions, part I: categories indexed by cyclic groups. The Ramanujan Journal.

[TaWe18] Tarrago, Pierre and Weber, Moritz, February 2018. The classification of tensor categories of two-colored non-crossing partitions. Journal of Combinatorial Theory, Series A, 154, pp.464–506.

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Color sum

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