A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.
1, 1, 0, 0, 2, 0, 0, 3, 3, 0, 0, 4, 16, 4, 0, 0, 5, 55, 55, 5, 0, 0, 6, 156, 396, 156, 6, 0, 0, 7, 399, 2114, 2114, 399, 7, 0, 0, 8, 960, 9528, 19328, 9528, 960, 8, 0, 0, 9, 2223, 38637, 140571, 140571, 38637, 2223, 9, 0, 0, 10, 5020, 146080, 882340, 1561900, 882340, 146080, 5020, 10, 0
Offset: 0
Examples
Triangle begins: 1; 1, 0; 0, 2, 0; 0, 3, 3, 0; 0, 4, 16, 4, 0; 0, 5, 55, 55, 5, 0; 0, 6, 156, 396, 156, 6, 0; 0, 7, 399, 2114, 2114, 399, 7, 0; 0, 8, 960, 9528, 19328, 9528, 960, 8, 0; ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Crossrefs
Programs
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PARI
T(n, k) = {if(n==0, k==0, n*sum(j=0, k, (-1)^j * (k-j)^(n-1) * binomial(n, j)))}
Formula
T(n, k) = n*A008292(n-1, k) for n > 1.
T(n, k) = T(n, n-k) for n > 1.
T(n, k) = n*Sum_{j=0..k} (-1)^j * (k-j)^(n-1) * binomial(n, j) for n > 0.