A112091 Number of idempotent order-preserving partial transformations (of an n-element chain).
1, 2, 6, 21, 76, 276, 1001, 3626, 13126, 47501, 171876, 621876, 2250001, 8140626, 29453126, 106562501, 385546876, 1394921876, 5046875001, 18259765626, 66064453126, 239023437501, 864794921876, 3128857421876, 11320312500001
Offset: 0
Examples
a(2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 2-element chain), namely: the empty map, (1)->(1), (2)->(2), (1,2)->(1,1), (1,2)->(1,2), (1,2)->(2,2); the mappings are coordinate-wise.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- D. Callan and T. Mansour, Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns, arXiv:1705.00933 [math.CO] (2017), Table 2 No 200 (offset 1 then).
- A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
- A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
- Index entries for linear recurrences with constant coefficients, signature (6,-10,5).
Programs
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Magma
[ n eq 1 select 1 else n eq 2 select 2 else n eq 3 select 6 else 6*Self(n-1)-10*Self(n-2)+ 5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 21 2011
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Mathematica
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==1+5(a[n-1]-a[n-2])},a[n], {n,30}] (* or *) LinearRecurrence[{6,-10,5},{1,2,6},31] (* Harvey P. Dale, Aug 20 2011 *) -
PARI
Vec((2*x-1)^2/(1-x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Aug 21 2011
Formula
a(n) = ((sqrt(5))^(n - 1))*(((sqrt(5) + 1)/2)^n - ((sqrt(5) - 1)/2)^n) + 1. [corrected by Jason Yuen, Sep 06 2024]
a(n) = 1 + 5*(a(n-1) - a(n-2)), a(0) = 1, a(1) = 2.
G.f.: (1 - 2*x)^2/((1 - x)*(1 - 5*x + 5*x^2)). Convolution of A081567 with the sequence 1, -1, -1, -1 (-1 continued). - R. J. Mathar, Sep 06 2008
a(n) = 1 + A030191(n-1). - R. J. Mathar, Jun 20 2011
a(n) = 6*a(n-1) - 10*a(n-2) + 5*a(n-3); a(0) = 1, a(1) = 2, a(2) = 6. - Harvey P. Dale, Aug 20 2011
E.g.f.: exp(x) + (exp((5 + sqrt(5))*x/2) - exp((5 - sqrt(5))*x/2))/sqrt(5). - Franck Maminirina Ramaharo, Nov 09 2018