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 A110858 Triangle read by rows: number of order-preserving partial transformations (of an n-element chain) of width and waist both equal to r (width(alpha) = |Dom(alpha)| and waist(alpha) = max(Im(alpha)). 1
 1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 20, 1, 5, 20, 60, 100, 70, 1, 6, 30, 120, 300, 420, 252, 1, 7, 42, 210, 700, 1470, 1764, 924, 1, 8, 56, 336, 1400, 3920, 7056, 7392, 3432, 1, 9, 72, 504, 2520, 8820, 21168, 33264, 30888, 12870, 1, 10, 90, 720, 4200, 17640, 52920, 110880, 154440, 128700, 48620 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Whenever the formula generates a zero, it is changed to a one in the sequence. - Harvey P. Dale, Sep 06 2013 LINKS Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359. Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8 FORMULA G(n,k) = binomial(n,k)*binomial(2*k-2,k-1), n >= k > 0. EXAMPLE G(3,2)=6 because there are exactly 6 order-preserving partial transformations (on a 3-element chain) of both width and waist equal to 2, namely: (1,2)->(1,2),(1,2)->(2,2),(1,3)->(1,2),(1,3)->(2,2),(2,3)->(1,2),(2,3)->(2,2) 1; 1,1; 1,2,2; 1,3,6,6; 1,4,12,24,20; 1,5,20,60,100,70; 1,6,30,120,300,420,252; 1,7,42,210,700,1470,1764,924; 1,8,56,336,1400,3920,7056,7392,3432; MAPLE A110858 := proc(n, k) if k = 0 then 1; else binomial(n, k)*binomial(2*k-2, k-1) ; end if; end proc: # R. J. Mathar, Jun 20 2011 MATHEMATICA Flatten[Table[Binomial[n, k]Binomial[2k-2, k-1], {n, 0, 10}, {k, 0, n}]/.{0->1}] (* Harvey P. Dale, Sep 04 2013 *) CROSSREFS Sequence in context: A107111 A082037 A163649 * A008279 A239572 A056043 Adjacent sequences:  A110855 A110856 A110857 * A110859 A110860 A110861 KEYWORD nonn,tabl AUTHOR Abdullahi Umar, Aug 25 2008 STATUS approved

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Last modified May 25 08:26 EDT 2020. Contains 334585 sequences. (Running on oeis4.)