### Abstract

A *h* *y* *p* *e* *r* *g* *r* *a* *p* *h* *H*={*E* _{1},*E* _{2},...,*E* _{ n }} is a set of sets; the sets *E* _{ i } are called the *e* *d* *g* *e* *s*, and the elements of *V*(*H*)≡U^{ n } _{ i=1} *E* _{ i } the *v* *e* *r* *t* *i* *c* *e* *s* *o* *f* *H*(if each edge contains only two vertices, we have an ordinary graph). Hypergraphs and *c* *u* *m* *u* *l* *a* *n* *t* *s* over hypergraphs occur in various statistical mechanical problems. A cumulant over *H* involves a sum over partitions of the set *H*, whence the motivation for studying partitions of hypergraphs and related notions. Writing *E* _{ i } ‐ ‐ ‐ ‐^{ k‐(v)} *E* _{ j } if there exist at least *k*∈N vertex‐disjoint paths between the edges *E* _{ i } and *E* _{ j }, we show that ‐ ‐ ‐ ‐^{ k‐(v)} is an equivalence relation on *H*, and denote by *P*̂_{ k } *H* the partition of *H* into its (‐ ‐ ‐ ‐^{ k‐(v)}) equivalence classes. *H* is said to be *k*‐(*v*) *c* *o* *n* *n* *e* *c* *t* *e* *d* if every pair of edges is in the relation ‐ ‐ ‐ ‐^{ k‐(v)}; we show that there exists a *c* *o* *a* *r* *s* *e* *s* *t* partition of *H* into *k*‐(*v*) connected blocks, and denote it by *P*̂^{′} _{ k } *H*. Given a partition *P*(*H*)={*H* _{1},*H* _{2},...,*H* _{ p }} of *H*, we denote, for any real number κ, σ_{κ}[*P*(*H*)]≡∑^{ p } _{ j=1} **(**‖*V*(*H* _{ j })‖ −κ**)**, σ_{κ}[*P*(*H*)] ≡∑^{ p } _{ j=1}(‖*V*(*H* _{ j })‖ −κmk*P*̂_{1} *H* _{ j }‖**)**, where ‖*V*(*H* _{ j })‖ is the number of vertices and ‖*P*̂_{1} *H* _{ j }‖ the number of connected components of *H* _{ j }. We introduce ‘‘subcumulants’’ over *H*, which involve only partitions of *H* for which σ^{′} _{κ} has the same value. The subcumulants corresponding to *m* *i* *n* *i* *m* *u* *m* values of σ_{κ}, κ∈R, are of practical interest (especially the cases κ=0 and 1). We accordingly study the set π^{′0} _{κ}(*H*) of partitions which minimize σ_{κ}; this is simply related to the set π^{0} _{κ}(*H*) of partitions which minimize σ_{κ}. We show that *P*(*H*)≤*P*̂^{′} _{ k } *H* ≤*P*̂_{ k } *H* for all *P*(*H*)∈π^{0} _{ k }(*H*) (where ≤ signifies ‘‘is a subpartition of’’), that *P*≤*P*′ for every *P*∈π^{0} _{ k }(*H*) and *P*′∈π^{0} _{κ−ε}(*H*), ε>0, and that π^{0} _{κ}(*H*) is a sublattice of the (≤) lattice of all partitions of *H*. The sets π^{0} _{κ}(*H*) and π^{0} _{κ}(*H*), and the corresponding minimal values of σ_{κ} and σ^{′} _{κ}, are explicitly determined, for any hypergraph if κ≤1, and for some special types of hypergraphs if κ>1. We introduce a new notion of connectedness for hypergraphs, the σ *c* *o* *n* *n* *e* *c* *t* *i* *v* *i* *t* *y* Σ(*H*)≡Max{κ,{*H*}∈π^{0} _{κ}(*H*)}, and relate it to the *v* *e* *r* *t* *e* *x* *c* *o* *n* *n* *e* *c* *t* *i* *v* *i* *t* *y* *k* _{ v }(*H*)≡Max{*k*, *H* is *k*‐(*v*)connected}; we have, in particular, Σ(*H*)≤*k* _{ v }(*H*) (Σ=*k* _{ v } if *k* _{ v }=0 or 1).

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