Abstract

We write$\mathcal {S}_n(A)$for the set of permutations of a setAwithnnon-fixed points and$\mathrm {{seq}}^{1-1}_n(A)$for the set of one-to-one sequences of elements ofAwith lengthnwherenis a natural number greater than$1$. With the Axiom of Choice,$|\mathcal {S}_n(A)|$and$|\mathrm {{seq}}^{1-1}_n(A)|$are equal for all infinite setsA. Among our results, we show, in ZF, that$|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$for any infinite setAif${\mathrm {AC}}_{\leq n}$is assumed and this assumption cannot be removed. In the other direction, we show that$|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$for any infinite setAand the subscript$n+1$cannot be reduced ton. Moreover, we also show that “$|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$for any infinite setA” is not provable in ZF.