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A344639
Array read by ascending antidiagonals: A(n, k) is the number of (n, k)-poly-Cauchy permutations.
2
1, 1, 1, 2, 2, 1, 6, 5, 4, 1, 24, 17, 13, 8, 1, 120, 74, 51, 35, 16, 1, 720, 394, 244, 161, 97, 32, 1, 5040, 2484, 1392, 854, 531, 275, 64, 1, 40320, 18108, 9260, 5248, 3148, 1817, 793, 128, 1, 362880, 149904, 70508, 36966, 20940, 12134, 6411, 2315, 256, 1
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
Cf. A000012 (n = 0), A000079 (n = 1), A000142 (k = 0), A000774 (k = 1), A007318, A007689 (n = 2), A008275, A008277, A081048, A192563 (diagonal), A223899 (k = 2), A223901 (k = 3), A223902 (k = 4), A223904 (k = 5), A344640 (antidiagonal sums).
Sequence in context: A127743 A125278 A134558 * A230420 A137381 A109316
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 25 2021
STATUS
approved