cp's OEIS Frontend

A261762 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, 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.

%I A261762 #39 Apr 13 2017 04:25:29
%S A261762 1,1,1,1,1,4,1,1,10,18,1,1,46,78,108,1,1,166,486,636,780,1,1,856,3096,
%T A261762 4896,5760,6600,1,1,3844,21204,40104,52200,58080,63840,1,1,21820,
%U A261762 167868,363168,508320,602400,648480,693840,1,1,114076,1370268,3490848,5450400,6720480
%N A261762 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, 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.
%C A261762 The OEIS values correct the values b(n,k) in the Laradji-Umar Table 2.1 in column k=2. Note that the row sums (meaning: sums up to the diagonal of the triangle) in Table 2.1 in the article are also incorrect.
%C A261762 There were typos in the column (k=2) of the original article. The entry 94 should be 166 and the entry 784 should be 856, which have been corrected. Unlike most triangles the off-diagonal terms are not 0 because T(n, n)= T(n, n+k) for all nonnegative k which is obvious from the definition.
%H A261762 A. Laradji and A. Umar, <a href="http://www.researchgate.net/profile/Abdullahi_Umar/publication/261689545_On_the_number_of_subpermutations_with_a_fixed_orbit_size/links/53e0a3f00cf24f90ff60849b.pdf">On the number of subpermutations with fixed orbit size</a>, Ars Combinatoria, 109 (2013), 447-460.
%F A261762 T(n,k) = T(n-1,k) + 3(n-1)T(n-2,k) + ... +(k+1)(n-1)(n-2)...(n-k+1)T(n-k,k) if k<=n.
%F A261762 T(n,k) = T(n,n) if k>n, not part of the triangle.
%F A261762 T(n,0) = T(n,1) = 1.
%F A261762 T(n,n) = A144085(n). (Diagonal)
%F A261762 G.f.: exp(x+(3x^2)/2+ ... +((k+1)x^k)/k).
%e A261762 T(3,2) = 10 because there are 10 subpermutations on {1,2,3}, each of whose orbit is of size at most 2, and without fixed points, namely: Empty map, (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 A261762 Triangle starts:
%e A261762 1;
%e A261762 1, 1;
%e A261762 1, 1, 4;
%e A261762 1, 1, 10, 18;
%e A261762 1, 1, 46, 78, 108;
%e A261762 1, 1, 166, 486, 636, 780;
%e A261762 ...
%p A261762 A261762 := proc(n,k)
%p A261762     if k = 0 then
%p A261762         1;
%p A261762     else
%p A261762         if k < 1 then
%p A261762             g := 1;
%p A261762         elif k < 2 then
%p A261762             g := exp(x) ;
%p A261762         else
%p A261762             g := exp(x+add((j+1)*x^j/j,j=2..k)) ;
%p A261762         fi;
%p A261762         coeftayl(g,x=0,n) *n! ;
%p A261762     end if;
%p A261762 end proc:
%p A261762 seq(seq( A261762(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Nov 04 2015
%t A261762 T[n_, k_] := SeriesCoefficient[ Exp[ x + Sum[ (j+1)*x^j/j, {j, 2, k}]], {x, 0, n}] * n!; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 13 2017 *)
%Y A261762 Cf. A157400, A261763, A261764, A261765, A261766, A261767.
%K A261762 nonn,tabl
%O A261762 0,6
%A A261762 _Samira Stitou_, Sep 21 2015
%E A261762 More terms from _Alois P. Heinz_, Oct 07 2015