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%I #20 Nov 04 2015 19:35:27
%S 1,1,0,1,0,3,1,0,9,8,1,0,45,32,30,1,0,165,320,150,144,1,0,855,2240,
%T 1800,864,840,1,0,3843,17360,18900,12096,5880,5760,1,0,21819,146048,
%U 195300,145152,94080,46080,45360,1,0,114075,1256192,2120580,1959552,1270080,829440,408240,403200
%N Triangle read by rows: T(n,k) is the number of subpermutations of an n-set, whose orbits are each of size at most k with at least one orbit of size exactly k, and without fixed points. Equivalently, T(n,k) is the number of partial derangements of an n-set each of whose orbits is of size at most k with at least one orbit of size exactly k, and without fixed points.
%C T(n,n) is A261766. Sum of rows is A144085.
%D A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.
%F T(n,k) = A261762(n,k) - A261762(n,k-1).
%e T(n,1) = 0 because there is no (partial) derangement with an orbit of size 1.
%e T(3,2) = 9 because there are 9 subpermutations on {1,2,3}, whose orbits are each of size at most 2 with at least one orbit of size exactly 2, and without fixed points, namely: (1 2 --> 2 1), (1 3 --> 3 1), (2 3 --> 3 2), (1-->2), (1-->3), (2-->1), (2-->3), (3-->1), (3-->2).
%e Triangle starts:
%e 1;
%e 1, 0;
%e 1, 0, 3;
%e 1, 0, 9, 8;
%e 1, 0, 45, 32, 30;
%e 1, 0, 165, 320, 150, 144;
%e 1, 0, 855, 2240, 1800, 864, 840;
%e ...
%Y Cf. A157400, A261762, A261763, A261764, A261766, A261767.
%K nonn,tabl
%O 0,6
%A _Samira Stitou_, Sep 21 2015
%E More terms from _Alois P. Heinz_, Nov 04 2015
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