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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.

%I A091441 #18 Nov 27 2017 02:35:04
%S A091441 1,2,2,6,8,6,24,36,36,24,120,192,216,192,120,720,1200,1440,1440,1200,
%T A091441 720,5040,8640,10800,11520,10800,8640,5040,40320,70560,90720,100800,
%U A091441 100800,90720,70560,40320,362880,645120,846720,967680,1008000,967680
%N 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.
%D A091441 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 114 (2.4.42).
%H A091441 Reinhard Zumkeller, <a href="/A091441/b091441.txt">Rows n = 1..125 of triangle, flattened</a>
%F A091441 Double e.g.f.: A(x, y) = Sum_{i, j>=0} (x^i*y^j/(i!*j!)) = (1-x)*(1-y)/(1-x-y).
%F A091441 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
%e A091441     1,    2,     6,     24,     120; ...
%e A091441     2,    8,    36,    192,    1200; ...
%e A091441     6,   36,   216,   1440,   10800; ...
%e A091441    24,  192,  1440,  11520,  100800; ...
%e A091441   120, 1200, 10800, 100800, 1008000; ...
%o A091441 (Haskell)
%o A091441 import Data.List (genericLength)
%o A091441 a091441 n k = a091441_tabl !! (n-1) !! (k-1)
%o A091441 a091441_row n = a091441_tabl !! (n-1)
%o A091441 a091441_tabl = iterate f [1] where
%o A091441    f xs = zipWith (+)
%o A091441      (zipWith (*) ([0] ++ xs) ks) (zipWith (*) (xs ++ [0]) (reverse ks))
%o A091441      where ks = [1 .. 1 + genericLength xs]
%o A091441 -- _Reinhard Zumkeller_, May 07 2013
%Y A091441 Cf. A008292.
%K A091441 nonn,tabl
%O A091441 1,2
%A A091441 _Christian G. Bower_, Jan 09 2004