The number of permutations of n elements that have exactly k inversions.

The number of permutations of n elements that have no inversions.

The number of permutations of n elements that have exactly k fixed points.

The number of permutations of n elements that have no fixed points.

$A(n, k) = A(n-1, k-1) + (n-k)A(n-1, k)$

$A(n, k) = A(n-1, k) + (n-k+1)A(n-1, k-1)$

$A(n, k) = A(n-1, k) + (n-k)A(n-1, k+1)$

$A(n, k) = A(n-1, k) + (n-k+1)A(n-1, k+1)$

$A(0, 0) = 1$

$A(1, 0) = 1$

$A(0, 1) = 1$

$A(1, 1) = 1$

$F(x) = \frac{1-x}{1-xe^x}$

$F(x) = \frac{1+x}{1+xe^x}$

$F(x) = \frac{1-x}{1+xe^x}$

$F(x) = \frac{1+x}{1-xe^x}$

$A(n, k) \sim \frac{1}{k!}n^k$

$A(n, k) \sim \frac{1}{k!}n^{k+1}$

$A(n, k) \sim \frac{1}{k!}n^{k-1}$

$A(n, k) \sim \frac{1}{k!}n^{k+2}$

$A(n, k) = \sum_{i=0}^k S(n, i)$

$A(n, k) = \sum_{i=0}^k S(n+1, i)$

$A(n, k) = \sum_{i=0}^k S(n, i+1)$

$A(n, k) = \sum_{i=0}^k S(n+1, i+1)$

$A(n, k) = L(n, k+1)$

$A(n, k) = L(n+1, k)$

$A(n, k) = L(n, k)$

$A(n, k) = L(n+1, k+1)$

$A(2n, n) = C_n$

$A(2n+1, n) = C_n$

$A(2n, n+1) = C_n$

$A(2n+1, n+1) = C_n$

$A(2n, n) = M_{2n}$

$A(2n+1, n) = M_{2n+1}$

$A(2n, n+1) = M_{2n}$

$A(2n+1, n+1) = M_{2n+1}$

$A(n, k) = N(n, k+1)$

$A(n, k) = N(n+1, k)$

$A(n, k) = N(n, k)$

$A(n, k) = N(n+1, k+1)$

$A(n, k) = D(n, k+1)$

$A(n, k) = D(n+1, k)$

$A(n, k) = D(n, k)$

$A(n, k) = D(n+1, k+1)$

$A(n, k) = S(n, k+1)$

$A(n, k) = S(n+1, k)$

$A(n, k) = S(n, k)$

$A(n, k) = S(n+1, k+1)$

$A(n, k) = G(n, k+1)$

$A(n, k) = G(n+1, k)$

$A(n, k) = G(n, k)$

$A(n, k) = G(n+1, k+1)$

$A(n, k) = B(n, k+1)$

$A(n, k) = B(n+1, k)$

$A(n, k) = B(n, k)$

$A(n, k) = B(n+1, k+1)$

$A(n, k) = F(n+k)$

$A(n, k) = F(n-k)$

$A(n, k) = F(n+k+1)$

$A(n, k) = F(n-k+1)$