A171996 A(n,k,m) is the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, called the (n,k)-th m-restrained Stirling numbers of the first kind, and denoted by mS_1(n,k). The sequence shows the case of m=3.
1, -1, 1, 2, -3, 1, 0, 11, -6, 1, 0, -20, 35, -10, 1, 0, 40, -135, 85, -15, 1, 0, 0, 490, -525, 175, -21, 1, 0, 0, -1120, 2905, -1540, 322, -28, 1, 0, 0, 2240, -12600, 11865, -3780, 546, -36, 1, 0, 0, 0, 47600, -76545, 38325, -8190, 870, -45, 1
Offset: 1
Examples
A(1,1,3)=1, A(1,2,3)=0, A(1,3,3)=0, A(1,4,3)=0, ... A(2,1,3)=-1, A(2,2,3)=1, A(2,3,3)=0, A(2,4,3)=0, ... A(3,1,3)=2, A(3,2,3)=-3, A(3,3,3)=1, A(3,4,3)=0, ... A(4,1,3)=0, A(4,2,3)=11, A(4,3,3)=-6, A(4,4,3)=1, ...
Links
- J. Y. Choi, Multi-restrained Stirling numbers
Programs
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PARI
T(n,k) = (-1)^(n-k)*n!*sum(j=0, k, binomial(j,n-k-j)*binomial(k,j)*3^(-n+k+j)*2^(n-k-2*j)/k!); \\ Jinyuan Wang, Dec 21 2019
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Sage
# uses[bell_matrix from A264428] # Adds a column 1, 0, 0, 0, ... at the left side of the triangle. bell_matrix(lambda n: [1,-1,2][n] if n < 3 else 0, 12) # Peter Luschny, Jan 19 2016
Formula
A(n,k,m) = Sum {(-1)^(n-k)*n!} /{1^{k_1}*2^{k_2}*...*m^{k_m}*(k_1!)*(k_2!)*...*(k_m!)}, where k_1 + 2*k_2 + ... + m*k_m = n and k_1 + k_2 + ... + k_m = k.
Recurrence A(n,k,m) = Sum_{i=1..m} (-1)^(i-1)*[n-1]_{i-1}*A(n-i,k-1,m).
A(n,k,m) = A(n-1,k-1,m) - (n-1)*A(n-1,k,m) - (-1)^m(n-1)*(n-2)*...*(n-m)*A(n-m-1,k-1,m).
Generating function f(t) = (1 + t - t^2/2 + t^3/3 + ... + (-1)^(m-1) t^m/m)^x, for an indeterminate x ===> the n-th derivative of f(t) at t=0, f^(n)(0) = Sum_{k=1..n} A(n,k,m)[x]_k, where [x]_k is the k-th falling factorial
T(n,k) = (-1)^(n-k)*n!*Sum_{j=0..k} C(j,n-k-j)*C(k,j)*3^(-n+k+j)*2^(n-k-2*j)/k!. - Vladimir Kruchinin, Oct 02 2019
Extensions
More terms from Peter Luschny, Jan 19 2016
Comments