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 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. 6
 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, 4896, 5760, 6600, 1, 1, 3844, 21204, 40104, 52200, 58080, 63840, 1, 1, 21820, 167868, 363168, 508320, 602400, 648480, 693840, 1, 1, 114076, 1370268, 3490848, 5450400, 6720480 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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. 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. LINKS A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460. FORMULA 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. T(n,k) = T(n,n) if k>n, not part of the triangle. T(n,0) = T(n,1) = 1. T(n,n) = A144085(n). (Diagonal) G.f.: exp(x+(3x^2)/2+ ... +((k+1)x^k)/k). EXAMPLE 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. Triangle starts: 1; 1, 1; 1, 1, 4; 1, 1, 10, 18; 1, 1, 46, 78, 108; 1, 1, 166, 486, 636, 780; ... MAPLE A261762 := proc(n, k)     if k = 0 then         1;     else         if k < 1 then             g := 1;         elif k < 2 then             g := exp(x) ;         else             g := exp(x+add((j+1)*x^j/j, j=2..k)) ;         fi;         coeftayl(g, x=0, n) *n! ;     end if; end proc: seq(seq( A261762(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Nov 04 2015 MATHEMATICA 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 *) CROSSREFS Cf. A157400, A261763, A261764, A261765, A261766, A261767. Sequence in context: A164366 A121692 A264614 * A225062 A145271 A232774 Adjacent sequences:  A261759 A261760 A261761 * A261763 A261764 A261765 KEYWORD nonn,tabl AUTHOR Samira Stitou, Sep 21 2015 EXTENSIONS More terms from Alois P. Heinz, Oct 07 2015 STATUS approved

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Last modified October 20 13:38 EDT 2021. Contains 348108 sequences. (Running on oeis4.)