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"... Submit only ONE copy of this form for each PI/PD and co-PI/PD identified on the proposal. The form(s) should be attached to the original proposal as specified in GPG Section II.C.a. Submission of this information is voluntary and is not a precondition of award. This information will ..."

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Submit only ONE copy of this form for each PI/PD and co-PI/PD identified on the proposal. The form(s) should be attached to the original proposal as specified in GPG Section II.C.a. Submission of this information is voluntary and is not a precondition of award. This information will

### INCIDENCE MATRIX AND COVER MATRIX OF NESTED INTERVAL ORDERS ∗

"... Abstract. For any poset P, its incidence matrix and its cover matrix C are the P ×P (0,1) matrices such that (x,y) = 1 if any only if x is less than y in P and C(x,y) = 1 if any only if x is covered by y in P. It is shown that and C are conjugate to each other in the incidence algebra of P over a ..."

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Abstract. For any poset P, its incidence matrix and its cover matrix C are the P ×P (0,1) matrices such that (x,y) = 1 if any only if x is less than y in P and C(x,y) = 1 if any only if x is covered by y in P. It is shown that and C are conjugate to each other in the incidence algebra of P over a field of characteristic 0 provided P is the nested interval order. In particular, when P is the Bruhat order of a dihedral group, which consists of a special family of nested intervals, and C turn out to be conjugate in the incidence algebra over every field. Moreover, and C are proved to be conjugate in the incidence algebra over every field when P is the weak order of a dihedral group. Many relevant problems and observations are also presented in this note. Key words. Hierarchy, Jordan canonical form, Rank, Strict incidence algebra. AMS subject classifications. 06A07, 06A11, 15A21, 15A24, 20F55. 1. Poset and its incidence algebra. A (finite) partially ordered set, also known as a (finite) poset [33, p. 97], is a finite set P together with a binary relation ≤P, which is often denoted ≤ if there is no confusion, such that: • for all x ∈ P,x ≤ x (reflexivity); • if x ≤ y and y ≤ x, then x = y (antisymmetry);