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Let O[n] and C[n] be the semigroup of all order-preserving and all order-preserving and regressive transformations on the set [n] = {1, . . . , n}, respectively. In this paper, we give a pattern on [n]. Combinatorial properties on subsemigroups of O[n] with restricted range to the pattern are investigated. AMS Subject Classification: 20M20, 11B39

COMBINATORIAL RESULTS FOR SEMIGROUPS OF ORDER-PRESERVING FULL TRANSFORMATIONS WITH PATTERN OF RANGE