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A344639 Array read by ascending antidiagonals: A(n, k) is the number of (n, k)-poly-Cauchy permutations.

%I A344639 #35 Apr 15 2025 08:26:25
%S A344639 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,
%T A344639 97,32,1,5040,2484,1392,854,531,275,64,1,40320,18108,9260,5248,3148,
%U A344639 1817,793,128,1,362880,149904,70508,36966,20940,12134,6411,2315,256,1
%N A344639 Array read by ascending antidiagonals: A(n, k) is the number of (n, k)-poly-Cauchy permutations.
%C A344639 An (n, k)-poly-Cauchy permutation is a permutation which satisfies the properties listed by Bényi and Ramírez in Definition 1.
%H A344639 Beáta Bényi and José Luis Ramírez, <a href="https://arxiv.org/abs/2105.04791">Poly-Cauchy numbers -- the combinatorics behind</a>, arXiv:2105.04791 [math.CO], 2021.
%F A344639 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).
%F A344639 A(n, 0) = n! = A000142(n) (see Example 6 in Bényi and Ramírez).
%F A344639 A(1, k) = 2^k = A000079(k) (see Example 7 in Bényi and Ramírez).
%F A344639 A(2, k) = 2^k + 3^k = A007689(k) (see Example 8 in Bényi and Ramírez).
%F A344639 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).
%F A344639 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).
%F A344639 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).
%F A344639 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).
%F A344639 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).
%F A344639 From _Seiichi Manyama_, Apr 15 2025: (Start)
%F A344639 E.g.f. of column k: Sum_{j>=0} (j+1)^k * (-log(1-x))^j / j!.
%F A344639 E.g.f. of column k: (1/(1-x)) * Sum_{j=0..k} Stirling2(k+1,j+1) * (-log(1-x))^j. (End)
%e A344639 n\k|   0     1     2     3     4 ...
%e A344639 ---+----------------------------
%e A344639 0  |   1     1     1     1     1 ...
%e A344639 1  |   1     2     4     8    16 ...
%e A344639 2  |   2     5    13    35    97 ...
%e A344639 3  |   6    17    51   161   531 ...
%e A344639 4  |  24    74   244   854  3148 ...
%e A344639 ...
%t A344639 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}]]
%Y A344639 Rows n=0..2 give A000012, A000079, A007689.
%Y A344639 Columns k=0..5 give A000142, A000774, |A223899|, |A223901|, |A223902|, |A223904|.
%Y A344639 Main diagonal gives A192563.
%Y A344639 Antidiagonal sums give A344640.
%Y A344639 Cf. A007318, A008275, A008277, A081048.
%K A344639 nonn,tabl
%O A344639 0,4
%A A344639 _Stefano Spezia_, May 25 2021