Young Tableaux
Wouter Meeussen
eu000949 at pophost.eunet.be
Wed May 12 23:42:00 CEST 1999
At 15:14 12-05-99 -0400, Emeric Deutsch (RP)[M} wrote:
>Hi,
>By sheer coincidence, I am reading a paper where the
>sequence A000085 (1,1,2,4,10,26,76, ...) comes up
>(Duluck and Penaud; Discrete Math., 117, 1993, 89-105).
>Defined as number of involutions (i. e. self-inverse
>permutations) on n letters, one can easily derive
>
> floor(1/2 n)
> -----
> \ n!
> J := n -> ) ----------------
> / k
> ----- k! (n - 2 k)! 2
> k = 0
>(authors refer to Lucas' Theorie des Nombres, Vol. 1).
>Indeed, Maple gives
> 1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504,
> 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440,
>
>I couldn't resist the simple exercise of deriving the ordinary
>generating function G(z). It is defined by
>> deq:=z^3*diff(G(z),z)-(1-z-z^2)*G(z)+1 = 0;
>
> 3 /d \ 2
> deq := z |-- G(z)| - (1 - z - z ) G(z) + 1 = 0
> \dz /
>
> ic := G(0) = 1
>Maple gives the solution I couldn't decipher:
> 1
> sol := G(z) = signum(exp(- 1/2 ----)) infinity
> 2
> z
>However, with the "series" command
>> sol:=dsolve({deq,ic},G(z),series);
>we obtain
> 2 3 4 5 6 7
> sol := G(z) = 1 + z + 2 z + 4 z + 10 z + 26 z + 76 z + 232 z
>
> 8 9 10 11 12
> + 764 z + 2620 z + 9496 z + 35696 z + 140152 z +
>
> 13 14 15 16
> 568504 z + 2390480 z + 10349536 z + 46206736 z +
>
> 17 18 19 20
> 211799312 z + 997313824 z + 4809701440 z + O(z )
>
>Emeric
>
>
>
>
>
>
>
>
EIS A000085 has the closed form
Table[HypergeometricU[-(n/2), 1/2, -(1/2)] / (-(1/2))^(-(-n/2)),{n,12}]
{1,2,4,10,26,76,232,764,2620,9496,35696,140152}
in Mathematica 3.0 :
HypergeometricU[a,b,z] is defined as the confluent hypergeometric function
U(a,b,z).
The function U(a,b,z) has the integral representation
U(a, b, z) =
1/Gamma[a] Integral[E^(-z t)t^(a-1)(1+t)^(b-a-1) ,{t,0,Infinity}]
setting G[n_]:=HypergeometricU[-(n/2), 1/2, -(1/2)] / (-(1/2))^(-(-n/2))
and operating the differential equation z^3*D[G[z],z]-(1-z-z^2)*G[z]+1 on
it, hoping to get "=0"
leads to a long long wait, but no result sofar.
Doesn't Maple simplify the Sum[..] ?
wouter.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be
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