We delimit the boundary between decidability versus undecidability of the weak monadic second-order logic of one successor (WS1S) extended with linear cardinality constraints of the form |X ₁|+...+|X _r| < |Y ₁|+...+|Y _s|, where the X _is and Y _js range over finite subsets of natural numbers. Our decidability and undecidability results are based on an extension of the classic logic-automata connection using a novel automaton model based on Parikh maps.