A K-theoretic analogue of RSK insertion and the Knuth equivalence relations were introduced in [1, 2]. The resulting K-Knuth equivalence relations on words and increasing tableaux on [n] have prompted investigation into the equivalence classes of tableaux arising from these relations. Of particular interest are the tableaux that are unique in their class, which we refer to as unique rectification targets (URTs). In this paper we give several new families of URTs and a bound on the length of intermediate words connecting two K-Knuth equivalent words. In addition, we describe an algorithm to determine if two words are K-Knuth equivalent and to compute all K-Knuth equivalence classes of tableaux on [n].

In this paper we show that the spherical cap discrepancy of the point set given by centers of pixels in the HEALPix tessellation (short for Hierarchical, Equal Area and iso-Latitude Pixelation) of the unit 2-sphere is lower and upper bounded by order square root of the number of points, and compute explicit constants. This adds to the known collection of explicitly constructed sets whose discrepancy converges with order N −1/2 , matching the asymptotic order for i.i.d. random point sets. We describe the HEALPix framework in more detail and give explicit formulas for the boundaries and pixel centers. We then introduce the notion of an n-convex curve and prove an upper bound on how many fundamental domains are intersected by such curves, and in particular we show that boundaries of spherical caps have this property. Lastly, we mention briefly that a jittered sampling technique works in the HEALPix framework as well.

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