Hatsuyume (初夢) is the Japanese word for the first dream had in the new year. Traditionally, the contents of the dream would foretell the luck of the dreamer in the ensuing year. In Japan, the night of December 31 was often passed without sleeping, thus the hatsuyume was often the dream seen the night of January 1. This explains why the day after the night of the "first dream" is known as Hatsuyume. This day is January 2 in the gregorian calendar but was different in the traditional Japanese calendar.
It is considered to be particularly good luck to dream of Mount Fuji, a hawk, and an eggplant. This belief has been in place since the early Edo period but there are various theories regarding the origins as to why this particular combination was considered to be auspicious. One theory suggests that this combination is lucky because Mount Fuji is Japan's highest mountain, the hawk is a clever and strong bird, and the word for eggplant (nasu or nasubi 茄子) suggests achieving something great (nasu 成す). Another theory suggests that this combination arose because Mount Fuji, falconry, and early eggplants were favorites of the shōgun Tokugawa Ieyasu.