Minimal Null Designs and a Density Theorem of Posets

Abstract

Classically, null designs were defined on the poset of subsets of a given finite set (boolean algebra). A null design is defined as a collection of weighted k-subsets such that the sum of the weights of ksubsets containing a t-subset is 0 for every t-subset, where 0 ≤ t < k ≤ n. Null designs are useful to understand designs or to construct new designs from a known one. They also deserve research as pure combinatorial objects. In particular, people have been interested in the minimum number of k-subsets of non-zero weight to make a non-zero null design, and the characterization of the null designs with the minimal number of k-subsets of non-zero weight, which we call minimal null designs. Minimal null designs were used to construct explicit bases of the space of null designs.The definition of null designs can be extended to any poset which has graded structure (ranked poset) as the boolean algebra does. In this paper, we prove general theorems on the structure of the null designs of finite ranked posets, which also yield a density theorem of finite ranked posets. We apply the theorems to two special posets-the boolean algebra and the generalized (q-analogue of) boolean algebra-to characterize the minimal null t-designs.

Polytopes of roots of type AN

“…The same argument excludes the case k £ B, I £ A also. Therefore, the vertices of p/,j t (F 7 + Jt ) n gi,j 2 Equation (1) is verified because of the following well known equation: for fixed integers k,l,m, REMARK 1. The 7i-cycle (12... n) is a Coxeter element of S n .…”

“…D EXAMPLE 2. Let n = 6, i = 3, j = 3, A = (1,3,2), and let Np (miiin2im3 , be the number of paths from (0,0) to (2,2) which exceed P(mi,m2,m 3 )-Then the number of paths from (0,0) to (2,2) ,3)) = 3 + 3 = since P (1 , 3 ,2) = ENNE and ^(2,1,3) = NEEN.…”

“…Notice that P n is a Young orbit polytope corresponding to the partition (n -1,1), which was introduced as a framework for many combinatorial optimisation problems [8]. Moreover, the set of roots of type A n _i is the set of minimal null designs of a certain type [2]. Hence, it is worth characterising the faces of P n .…”

“…We refer to [1] and [10] for detailed information on convex polytopes, while we give r i some basic definitions. A (convex) polytope is the convex hull Conv(.ft') = i ^2 A,u; : S. Cho [2] Y2 A,; = 1, A, ^ 0 > of a finite set K -{ u i , . .…”

“…, 1), hence (n -1)! times the volume of Pfj is the number of paths from (0,0) to (i -l,n -i -1), which is ( n r2 x ).…”