  The problem of efficiently characterizing degree sequences of simple
hypergraphs is a fundamental long-standing open problem in Graph Theory.
Several results are known for restricted versions of this problem. This paper
adds to the list of sufficient conditions for a degree sequence to be {\em
hypergraphic}. This paper proves a combinatorial lemma about cyclically
permuting the columns of a binary table with length $n$ binary sequences as
rows. We prove that for any set of cyclic permutations acting on its columns,
the resulting table has all of its $2^n$ rows distinct. Using this property, we
first define a subset {\em cyclic hyper degrees} of hypergraphic sequences and
show that they admit a polynomial time recognition algorithm. Next, we prove
that there are at least $2^{\frac{(n-1)(n-2)}{2}}$ {\em cyclic hyper degrees},
which also serves as a lower bound on the number of {\em hypergraphic}
sequences. The {\em cyclic hyper degrees} also enjoy a structural
characterization, they are the integral points contained in the union of some
$n$-dimensional rectangles.
