부산대학교 도서관

For impartial games Γ and Γ′, the Sprague-Grundy function of the disjunctive sum Γ + Γ′ is equal to the Nim-sum of their Sprague-Grundy functions. In this paper, we introduce p-calm subtraction games and show that for p-calm subtraction games Γ and Γ′, the Sprague-Grundy function of a p-saturation of Γ+Γ′ is equal to the p-Nim-sum of the Sprague-Grundy functions of their p-saturations. Here a p-Nim-sum is the result of addition without carrying in base p, and a p-saturation of Γ is an impartial game obtained from Γ by adding some moves. It will turn out that Nim and Welter’s game are p-calm. Further, using the p-calmness of Welter’s game, we generalize a relation between Welter’s game and representations of symmetric groups to disjunctive sums of Welter’s games and representations of generalized symmetric groups; this result is described combinatorially in terms of Young diagrams.