People with Down syndrome are often seen as "happy, loving, and sweet". As anybody who knows actual people with Down syndrome, they have a full range of emotions and dispositions. So where does the stereotype come from?
My husband suggested the following hypothesis:
Because children with Down syndrome are disproportionately aborted (or, in some cultures, institutionalized), the parents who choose to keep and raise these children are probably endowed with greater compassion and kindness than average. And this parenting translates into children that grow up to be compassionate, kind people, regardless of genetics.
This is not a complete explanation. Children with other disabilities are also vulnerable to abortion and abandonment, and yet children with spina bifida, blindness, or physical deformity do not have this reputation. I think there is a feedback loop, though. Children with Down syndrome have characteristics which elicit a different kind of parenting. And this parenting does seem to encourage the stereotype.