# 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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