The computational complexity of rules for the character table of Sn

The Murnaghan–Nakayama rule is the classical formula for computing the character table of Sn. Y. Roichman (Adv. Math. 129 (1997) 25) has recently discovered a rule for the Kazhdan–Lusztig characters of qHecke algebras of type A, which can also be used for the character table of Sn. For each of the two rules, we give an algorithm for computing entries in the character table of Sn. We then analyze the computational complexity of the two algorithms, and in the case of characters indexed by partitions in the (k,ℓ) hook, compare their complexities to each other. It turns out that the algorithm based on the Murnaghan–Nakayama rule requires far less operations than the other algorithm. We note the algorithms’ complexities’ relation to two enumeration problems of Young diagrams and Young tableaux.