<p>This paper gives an algebraic presentation of an algebra called the fused permutations algebra in the one-boundary case. It is obtained through a detailed study of the cyclotomic degenerate affine Hecke algebra. In particular, we prove that the fused permutations algebra is a quotient of the cyclotomic degenerate affine Hecke algebra, and we also describe a basis combinatorially in terms of signed permutations with avoiding patterns. In order to understand this quotient, we study the primitive idempotents of this cyclotomic degenerate affine Hecke algebra.</p>