n ⌈ log 2 ⁡ ( n ! ) ⌉ {\displaystyle \left\lceil \log _{2}(n!)\right\rceil } Minimum 1 0 0 2 1 1 3 3 3 4 5 5 5 7 7 6 10 10 7 13 13 8 16 16 9 19 19 10 22 22 11 26 26 12 29 30[2][3] 13 33 34[4][5][6] 14 37 38[6] 15 41 42[7][8][9] 16 45 45 or 46[10] 17 49 49 or 50 18 53 53 or 54 19 57 58[9] 20 62 62 21 66 66 22 70 71[6] n ⌈ log 2 ⁡ ( n ! ) ⌉ {\displaystyle \left\lceil \log _{2}(n!)\right\rceil } n log 2 ⁡ n − n ln ⁡ 2 {\displaystyle n\log _{2}n-{\frac {n}{\ln 2}}} 10 22 19 100 525 521 1 000 8 530 8 524 10 000 118 459 118 451 100 000 1 516 705 1 516 695 1 000 000 18 488 885 18 488 874 Above: A comparison of the lower bound ⌈ log 2 ⁡ ( n ! ) ⌉ {\displaystyle \left\lceil \log _{2}(n!)\right\rceil } to the actual minimum number of comparisons (from OEIS: A036604) required to sort a list of n items (for the worst case). Below: Using Stirling's approximation, this lower bound is well-approximated by n log 2 ⁡ n − n ln ⁡ 2 {\displaystyle n\log _{2}n-{\frac {n}{\ln 2}}} .