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I have never seen this word in either Hindi or Sanskrit. In Sanskrit, aya is related to the root i which means to rise or move, and means to move. By extension it could mean a move in a game, or could mean a die (as in dice) or luck. aMsha is probably related to a root that meant to attain or reach, and as you pointed out, means a share or a part. Grammatically, they can form a compound word ayAMsha meaning part of any of the meanings of aya, but I have never seen it used.