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

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

D. Callan, 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).

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

Index entries for linear recurrences with constant coefficients, signature (6,-10,5).

Formula

a(n) = ((sqrt(5))^(n - 1))*(((sqrt(5) + 1)/2)^n - ((sqrt(5) - 1)/2)^n)); 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

Example

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.

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 *)

Prog

(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
(PARI) Vec((2*x-1)^2/(1-x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Aug 21 2011

Crossrefs

Sequence in context: A294819 A294820 A116782 * A108146 A116798 A279560

Adjacent sequences: A112088 A112089 A112090 * A112092 A112093 A112094

Keyword

nonn,easy

Author

Abdullahi Umar, Aug 25 2008

Status

approved