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 A144089 T(n,k) is the number of partial bijections (or subpermutations) of an n-element set of height k (height(alpha) = |Im(alpha)|) and without fixed points. 1
 1, 1, 0, 1, 2, 1, 1, 6, 9, 2, 1, 12, 42, 44, 9, 1, 20, 130, 320, 265, 44, 1, 30, 315, 1420, 2715, 1854, 265, 1, 42, 651, 4690, 16275, 25494, 14833, 1854, 1, 56, 1204, 12712, 70070, 198184, 263284, 133496, 14833, 1, 72, 2052, 29904, 240534, 1076544, 2573508 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Rows also give coefficients of the matching-generating polynomial of the n-crown graph. - Eric W. Weisstein May 19 2017 LINKS Table of n, a(n) for n=0..51. A. Laradji and A. Umar, Combinatorial results for the symmetric inverse semigroup, Semigroup Forum 75, (2007), 221-236. Eric Weisstein's World of Mathematics, Crown Graph Eric Weisstein's World of Mathematics, Matching-Generating Polynomial FORMULA T(n,k) = (n!/(n-k)!)*Sum_{m=0..k}(-1^m/m!)*binomial(n-m,k-m). T(n,n-1) = A000166(n+1) and T(n,n) = A000166(n). E.g.f.: exp(log(1/(1-y*x))-y*x)*exp(x/(1 - y*x)). - Geoffrey Critzer, Feb 18 2022 EXAMPLE T(3,2) = 9 because there are exactly 9 partial bijections (on a 3-element set) without fixed points and of height 2, namely: (1,2)->(2,1), (1,2)->(2,3), (1,2)->(3,1), (1,3)->(2,1), (1,3)->(3,1), (1,3)->(3,2), (2,3)->(1,2), (2,3)->(3,1), (2,3)->(3,2),- the mappings are coordinate-wise. Triangle starts: 1; 1, 0; 1, 2, 1; 1, 6, 9, 2; 1, 12, 42, 44, 9; 1, 20, 130, 320, 265, 44; MATHEMATICA t[n_, k_] := n!^2*Hypergeometric1F1[-k, -n, -1]/(k!*(n-k)!^2); Flatten[ Table[ t[n, k], {n, 0, 7}, {k, 0, n}]] (* Jean-François Alcover, Oct 13 2011 *) CoefficientList[Table[x^n n! Sum[(-1)^k/k! LaguerreL[n - k, -1/x], {k, 0, n}], {n, 2, 10}], x] // Flatten (* Eric W. Weisstein, May 19 2017 *) PROG (Sage) def A144089_triangle(dim): # computes rows in reversed order M = matrix(ZZ, dim, dim) for n in (0..dim-1): M[n, n] = 1 for n in (1..dim-1): for k in (0..n-1): M[n, k] = M[n-1, k-1]+(2*k)*M[n-1, k]+(k+1)^2*M[n-1, k+1] return M A144089_triangle(9) # Peter Luschny, Sep 19 2012 CROSSREFS Row sums give A144085. Cf. A000166. Sequence in context: A137376 A039761 A196073 * A172107 A349226 A165891 Adjacent sequences: A144086 A144087 A144088 * A144090 A144091 A144092 KEYWORD nice,nonn,tabl AUTHOR Abdullahi Umar, Sep 11 2008 STATUS approved

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Last modified August 15 15:45 EDT 2024. Contains 375173 sequences. (Running on oeis4.)