The generating function for the, 1, -dim. Boolean latticee in q is:

1/(q-1)/(q^2-1)
                            The first 40 terms are

[1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12,

    13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21]

      The generating function for the, 2, -dim. Boolean latticee in q is:

1/(q^2-1)^2/(q^3-1)/(q-1)
                            The first 40 terms are

[1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166,

    192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762,

    819, 889, 952, 1029, 1099, 1183]

      The generating function for the, 3, -dim. Boolean latticee in q is:

(q^16+2*q^14+2*q^13+3*q^12+3*q^11+5*q^10+4*q^9+8*q^8+4*q^7+5*q^6+3*q^5+3*q^4+2*
q^3+2*q^2+1)/(q^2-1)/(q^3-1)/(q-1)/(q^4-1)/(q^6-1)/(q^7-1)/(q^8-1)/(q^5-1)
                            The first 40 terms are

[1, 1, 4, 7, 14, 23, 41, 63, 104, 152, 230, 327, 470, 647, 897, 1202, 1616,

    2117, 2775, 3566, 4580, 5787, 7301, 9092, 11298, 13885, 17028, 20688, 25076,

    30154, 36172, 43094, 51221, 60511, 71323, 83622, 97822, 113893, 132323,

    153083, 176740]

                                     3.112