#Mathematica, 59 48 bytes

Select[Range[#^#]-1,xMax[x~DigitCount~#]==1]&

Contains U+F4A1 "Private Use" character

#Explanation

Range[#^#]-1

Generate {1, 2, ..., n^n}. Subtract 1. (yields {0, 1, ..., n^n - 1})

xMax[x~DigitCount~#]==1

A Boolean function: True if each digit occurs at most once in base n.

Select[ ... ]

From the list {0, 1, ..., n^n - 1}, select ones that give True when the above Boolean function is applied.

59 byte version

Select[Range[#^#]-1,xDuplicateFreeQ[x~IntegerDigits~#]]&

Mathematica, 59 48 bytes

Select[Range[#^#]-1,xMax[x~DigitCount~#]==1]&

Contains U+F4A1 "Private Use" character

Explanation

Range[#^#]-1

Generate {1, 2, ..., n^n}. Subtract 1. (yields {0, 1, ..., n^n - 1})

xMax[x~DigitCount~#]==1

A Boolean function: True if each digit occurs at most once in base n.

Select[ ... ]

From the list {0, 1, ..., n^n - 1}, select ones that give True when the above Boolean function is applied.

59 byte version

Select[Range[#^#]-1,xDuplicateFreeQ[x~IntegerDigits~#]]&

#Mathematica, 59 bytes

Select[Range[#^#]-1,xDuplicateFreeQ[x~IntegerDigits~#]]&

#Mathematica, 59 48 bytes

Select[Range[#^#]-1,xMax[x~DigitCount~#]==1]&

Contains U+F4A1 "Private Use" character

#Explanation

Range[#^#]-1

Generate {1, 2, ..., n^n}. Subtract 1. (yields {0, 1, ..., n^n - 1})

xMax[x~DigitCount~#]==1

A Boolean function: True if each digit occurs at most once in base n.

Select[ ... ]

From the list {0, 1, ..., n^n - 1}, select ones that give True when the above Boolean function is applied.

59 byte version

Select[Range[#^#]-1,xDuplicateFreeQ[x~IntegerDigits~#]]&

#Mathematica, 57 bytes

Select[Range[#^#]-1,xDuplicateFreeQ[x~IntegerDigits~#]]&

#Mathematica, 59 bytes

Select[Range[#^#]-1,xDuplicateFreeQ[x~IntegerDigits~#]]&