Stirling Numbers of the Second Kind

$S(2, 1)$ represents the number of ways to partition a set of 2 elements into 1 nonempty subset. There is only one way to do this, so $S(2, 1) = 1$.

$S(3, 2)$ represents the number of ways to partition a set of 3 elements into 2 nonempty subsets. There are 3 ways to do this: {1, 2} and {3}, {1, 3} and {2}, and {2, 3} and {1}. Therefore, $S(3, 2) = 3$.

$S(4, 3)$ represents the number of ways to partition a set of 4 elements into 3 nonempty subsets. There are 6 ways to do this: {1, 2, 3} and {4}, {1, 2, 4} and {3}, {1, 3, 4} and {2}, {2, 3, 4} and {1}, {1, 2} and {3, 4}, and {1, 3} and {2, 4}. Therefore, $S(4, 3) = 6$.

$S(5, 4)$ represents the number of ways to partition a set of 5 elements into 4 nonempty subsets. There are 15 ways to do this. Therefore, $S(5, 4) = 15$.

$S(6, 5)$ represents the number of ways to partition a set of 6 elements into 5 nonempty subsets. There are 20 ways to do this. Therefore, $S(6, 5) = 20$.

$S(7, 6)$ represents the number of ways to partition a set of 7 elements into 6 nonempty subsets. There are 35 ways to do this. Therefore, $S(7, 6) = 35$.

$S(8, 7)$ represents the number of ways to partition a set of 8 elements into 7 nonempty subsets. There are 56 ways to do this. Therefore, $S(8, 7) = 56$.

$S(9, 8)$ represents the number of ways to partition a set of 9 elements into 8 nonempty subsets. There are 84 ways to do this. Therefore, $S(9, 8) = 84$.

$S(10, 9)$ represents the number of ways to partition a set of 10 elements into 9 nonempty subsets. There are 120 ways to do this. Therefore, $S(10, 9) = 120$.

$S(11, 10)$ represents the number of ways to partition a set of 11 elements into 10 nonempty subsets. There are 165 ways to do this. Therefore, $S(11, 10) = 165$.

$S(12, 11)$ represents the number of ways to partition a set of 12 elements into 11 nonempty subsets. There are 220 ways to do this. Therefore, $S(12, 11) = 220$.

$S(13, 12)$ represents the number of ways to partition a set of 13 elements into 12 nonempty subsets. There are 286 ways to do this. Therefore, $S(13, 12) = 286$.

$S(14, 13)$ represents the number of ways to partition a set of 14 elements into 13 nonempty subsets. There are 364 ways to do this. Therefore, $S(14, 13) = 364$.

$S(15, 14)$ represents the number of ways to partition a set of 15 elements into 14 nonempty subsets. There are 455 ways to do this. Therefore, $S(15, 14) = 455$.

$S(16, 15)$ represents the number of ways to partition a set of 16 elements into 15 nonempty subsets. There are 560 ways to do this. Therefore, $S(16, 15) = 560$.