A091441 Table (by antidiagonals) of permutations of two types of objects such that each cycle contains at least one object of each type. Each type of object is labeled from its own label set.
1, 2, 2, 6, 8, 6, 24, 36, 36, 24, 120, 192, 216, 192, 120, 720, 1200, 1440, 1440, 1200, 720, 5040, 8640, 10800, 11520, 10800, 8640, 5040, 40320, 70560, 90720, 100800, 100800, 90720, 70560, 40320, 362880, 645120, 846720, 967680, 1008000, 967680
Offset: 1
Examples
1, 2, 6, 24, 120; ...
2, 8, 36, 192, 1200; ...
6, 36, 216, 1440, 10800; ...
24, 192, 1440, 11520, 100800; ...
120, 1200, 10800, 100800, 1008000; ...
References
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 114 (2.4.42).
Links
- Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Crossrefs
Cf. A008292.
Programs
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Haskell
import Data.List (genericLength) a091441 n k = a091441_tabl !! (n-1) !! (k-1) a091441_row n = a091441_tabl !! (n-1) a091441_tabl = iterate f [1] where f xs = zipWith (+) (zipWith (*) ([0] ++ xs) ks) (zipWith (*) (xs ++ [0]) (reverse ks)) where ks = [1 .. 1 + genericLength xs] -- Reinhard Zumkeller, May 07 2013
Formula
Double e.g.f.: A(x, y) = Sum_{i, j>=0} (x^i*y^j/(i!*j!)) = (1-x)*(1-y)/(1-x-y).
T(n,k) = k * T(n-1,k-1) + (n-k+1) * T(n-1,k), T(1,1) = 1. - Reinhard Zumkeller, May 07 2013