A144091 - OEIS

%I #8 Apr 27 2016 12:24:18

%S 1,3,0,6,12,6,10,60,90,20,15,180,630,660,135,21,420,2730,6720,5565,

%T 924,28,840,8820,39760,76020,51912,7420,36,1512,23436,168840,585900,

%U 917784,533988,66744

%N T(n,k) is the number of partial bijections (or subpermutations) of an n-element set of height k (height(alpha) = |Im(alpha)|) and with exactly 2 fixed points

%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1007/s00233-007-0732-8">Combinatorial results for the symmetric inverse semigroup</a>, Semigroup Forum 75, (2007), 221-236.

%F T(n,k) = (n!/2!(n-k)!)sum(m=0,k-2,(-1^m/m!)C(n-2-m,k-2-m)).

%e T(4,2) = 6 because there are exactly 6 partial bijections (on a 4-element set) with exactly 2 fixed points and of height 2, namely: the 6 partial identities on 2-element subsets of the 4-element set.

%o (PARI) T(n,k) = (n!/2!*(n-k)!)*sum(m=0,k-2,((-1)^m/m!)*binomial(n-2-m,k-2-m))

%o for (n=2, 10, for (k=2, n, print1(T(n,k), ", "))) \\ _Michel Marcus_, Apr 27 2016

%Y Row sums are A144087.

%K nonn,tabl

%O 2,2

%A _Abdullahi Umar_, Sep 11 2008