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A144091 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.

%I A144091 #10 Jul 22 2025 06:11:33
%S A144091 1,3,0,6,12,6,10,60,90,20,15,180,630,660,135,21,420,2730,6720,5565,
%T A144091 924,28,840,8820,39760,76020,51912,7420,36,1512,23436,168840,585900,
%U A144091 917784,533988,66744
%N A144091 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 A144091 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 A144091 T(n,k) = (n!/2!(n-k)!)sum(m=0,k-2,(-1^m/m!)C(n-2-m,k-2-m)).
%e A144091 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 A144091 (PARI) T(n,k) = (n!/2!*(n-k)!)*sum(m=0,k-2,((-1)^m/m!)*binomial(n-2-m,k-2-m))
%o A144091 for (n=2, 10, for (k=2, n, print1(T(n,k), ", "))) \\ _Michel Marcus_, Apr 27 2016
%Y A144091 Row sums are A144087.
%K A144091 nonn,tabl
%O A144091 2,2
%A A144091 _Abdullahi Umar_, Sep 11 2008