A171998 - OEIS

A171998

In general, let A(n,k,m) denote the (n,k)-th entry of the inverse of the matrix consisting of the (n,k)-th m-restrained Stirling numbers of the second kind (-1)^(n-k) times the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, as the (n+1,k+1)-th entry. The sequence shows A(n,k,3), which is a lower triangular matrix, read by rows.

1, 1, 1, 1, 3, 1, -5, 7, 6, 1, -65, -15, 25, 10, 1, -455, -455, 0, 65, 15, 1, -1295, -4725, -1715, 140, 140, 21, 1

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OFFSET

1,5

COMMENTS

A(n,k,m) also can be expanded for nonpositive integers n and k using the m-restrained Stirling numbers of the first kind.

LINKS

Table of n, a(n) for n=1..28.

Ji Young Choi, Multi-restrained Stirling numbers, Ars Comb. 120 (2015), 113-127.

John Engbers, David Galvin, and Cliff Smyth, Restricted Stirling and Lah number matrices and their inverses, Journal of Combinatorial Theory, Series A, 161 (2019), 271-298.

FORMULA

A(n,k,m) = A(n-1,k-1,m) - Sum_{i=1..m-1} (-1)^{i}(k)...(k+i-1) A(n,k+i,m) A(n,k,m) = A(n-1,k-1,m) + k A(n-1,k,m) + (-1)^m k(k+1)...(k+m-1)A(n,k+m,m).

EXAMPLE

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) = 1, A(3,2,3) = 3, A(3,3,3) = 1, A(3,4,3) = 0, ...

A(4,1,3) = -5, A(4,2,3) = 7, A(4,3,3) = 6, A(4,4,3) = 1, ...

In other words, A(n,k,3) is the matrix

1 1

1 3 1

-5 7 6 1

...