OFFSET
0,4
COMMENTS
An (n, k)-poly-Cauchy permutation is a permutation which satisfies the properties listed by Bényi and Ramírez in Definition 1.
LINKS
Beáta Bényi and José Luis Ramírez, Poly-Cauchy numbers -- the combinatorics behind, arXiv:2105.04791 [math.CO], 2021.
FORMULA
A(n, k) = Sum_{m=0..n} abs(S1(n, m))*(m + 1)^k, where S1 indicates the signed Stirling numbers of first kind (see Theorem 5 in Bényi and Ramírez).
A(n, 0) = n! = A000142(n) (see Example 6 in Bényi and Ramírez).
A(1, k) = 2^k = A000079(k) (see Example 7 in Bényi and Ramírez).
A(2, k) = 2^k + 3^k = A007689(k) (see Example 8 in Bényi and Ramírez).
Sum_{m=0..n} (-1)^m*S2(n, m)*A(m, k) = (-1)^n*(n + 1)^k, where S2 indicates the Stirling numbers of the second kind (see Theorem 9 in Bényi and Ramírez).
A(n, k) = Sum_{j=0..k} j!*abs(S1(n+1, j+1))*S2(k+1, j+1) (see Theorem 14 in Bényi and Ramírez).
A(n, k) = (n - 1)*A(n-1, k) + Sum_{i=0..k} C(k, i)*A(n-1, k-i) for n > 0 (see Theorem 15 in Bényi and Ramírez).
A(n, k) = Sum_{i=0..n} Sum_{j=0..k} C(n-1, i)*i!*C(k, j)*A(n-1-i, k-j) for n > 0 (see Theorem 17 in Bényi and Ramírez).
A(n, k) = Sum_{m=0..n} Sum_{i=0..m} C(k-i, m-i)*S2(k, i)*abs(S1(n+1, m+1)) (see Theorem 18 in Bényi and Ramírez).
EXAMPLE
n\k| 0 1 2 3 4 ...
---+----------------------------
0 | 1 1 1 1 1 ...
1 | 1 2 4 8 16 ...
2 | 2 5 13 35 97 ...
3 | 6 17 51 161 531 ...
4 | 24 74 244 854 3148 ...
...
MATHEMATICA
A[n_, k_]:=Sum[Abs[StirlingS1[n, m]](m+1)^k, {m, 0, n}]; Flatten[Table[A[n-k, k], {n, 0, 9}, {k, 0, n}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 25 2021
STATUS
approved