As far as I know, humans have 23 pairs of chromosomes, each one which contains a particular amount of genes. But in the "last" pair, men have a XY pair chromosome, and women have a XX pair chromosome. Does the missing "leg" of the XY pair make men to have fewer genes than women, and if so, how many genes do each sex have?
It is true that the Y chromosome is shorter than the X chromosome and that there are more genes on the X chromosome.
Do men have fewer genes?
One could (mis)understand three things in the expression "number of genes".
Thanks to @GerardoFurtado for correcting my semantic in the comments.
1. Number of gene copies
From the statement that there are fewer genes on the Y chromosome, one can conclude that men have fewer genes copies than woman.
This is the intuition the OP seemed to have.
2. Number of genes
Men also have an X chromosome. So men have the standard genes present on the X chromosome (but they only have a single copy of it while women have two copies; btw you might be interested in dosage compensation).
Because women do not have a Y chromosome and because there are a number of genes on the Y chromosome that are not present on the X chromosome, men have genes that female don't have at all. Therefore women have fewer genes than men.
3. Number of alleles
There is not much reason to expect that one gender would be more heterozygote than the other at autosomes (=non sexual chromosomes). Some may hypothesize that women may have more heterozygosity than men if there is stronger selection among sperm than among ovules or things like that but let's not get down this complicated path.
One one hand women have more gene copies and therefore might experience more heterozygosity, one the other hand, men have more genes and would therefore eventually carry more alleles. I don't know which side wins!
Did you mean number of genes per cell or per individual?
So far I assumed you were interested about the number of genes (or gene copies) per cells but if you want to compare whole individuals than it is a different story!
Men are on average taller and therefore have more cells. Therefore if you compare the body-wide number of gene copies, women will have fewer gene copy on average (Thanks to @JM97 comment).
Note : the number of genes evolves.
For who is interested and can understand french, heres a link speaking, in a short way, about that, and about the eventual disappearence of the y chromosom in human genom:
And a link about the inactivation of one of the X chromosom in women genom (corpuscle of Barr) : http://www.snv.jussieu.fr/vie/dossiers/kx/kx.htm
As @GerardoFurtado noted in his comment, males have more genes since there are genes unique to the Y chromosome, like SRY. Of course, the fact that males only have a single X chromosome means that males will only have a single allele of each gene encoded by the X chromosome, while females have two alleles for each gene they have.
As Remi.b said, there are different definitions of what a gene is. I found some more sources, if I understand correctly, according to the table in this wikipedia article with Data source of Ensembl genome browser release 87 https://en.wikipedia.org/wiki/Human_genome, a male human would have 14600 "pseudogenes" and a female human would have 14800 "pseudogenes" in each cell, and according this another table https://en.wikipedia.org/wiki/Chromosome#cite_note-26 , a male human would have 21000 genes and a female woman would have 21750 genes in each cell
Note - this is a very simple way to present things, and don't have the ambition to be a biology thesis.
Men do not have fewer genes than women, but fewer allels (variations of a gene); the allels on the x-y chromosome pair are not necessarily the same on the x and on the y (for men) or second x (for women), but they are for a part of same kind - same gene. It's like if, for a part, the y chromosome was broken, missing a bar to be an x.
One example of this is daltonism, linked to the xy chromosome pair.
Let's say that the daltonism allel is a corrupted existing allel (a variation of a gene).
To be a daltonist, a woman must have the daltonism allel on both of his x chromosomes - the daltonism allel is not a dominant allel, and if there is one corrupted allel on one x chromosome, but a normal one on the other x, the woman won't be a daltonist.
But a man is a daltonist only if his X chromosome do have the "corrupted" allel, because of the missing part of the y chromosome (compared to an x one).
If a woman has both allels, one "good" and one "corrupted", she will transmit it or not to his child, depending on which X chromosome she will give to this child.
If the father is a daltonist, he won't transmit it to his child if this child is a boy (he gives him his y chromosome genes and allels, not the x ones, and the boy takes an x from the mother, so he is xy).
But if his child is a girl, the father will transmit it to her (the girl take an x from the two of his mother and the only x from the father, so she is xx)
So, a child, to be a daltonist, must have at least a daltonist allel from his mother, no matter if the woman was daltonist or not (one x allel only corrupted or both).
Then, if this child is a girl, and is daltonist, it implies his father was daltonist too.
If it's a boy, no matter if the father is daltonist or not, he will be a daltonist as soon as his mother transmitted him a corrupted allel.
If the father is a daltonist, and if the mother has no daltonism allel at all, he will although transmit the allel to his daughters, who, while not being daltonists, will be able to transmit it to their own child.
In the same case (father daltonist, mother with no daltonism allel at all), if the child is a boy, he won't be a daltonist and won't be able to transmit it to his own child, etc, etc.
Remember, this is a simple way to present things, and do not pretend to be at the top of today's science vision, and even more not the truth - above the fact I'm not a biology specialist, every scientific that deserves this name won't ever be claimed to say the truth, something reserved for religions - and that's why science and knowledge, in general as well as for each people in the same spirit state, progresses ...
Last word, concerning the numbers of genes, according to Wikipedia - but you know that Wikipedia is not always the best source ;) - a human male has an estimated amount of 21 000 genes, and a woman, something between 0 and 50 fewer genes (800 genes on the x chromosome, 50 on the y one, but some of these are commons to both. How many, I don't know ...).
Sorry for the multiple edits, I try to give my 2 cents on this question after 2 days and a night without sleeping
Some of the information contained in this post requires additional references. Please edit to add citations to reliable sources that support the assertions made here. Unsourced material may be disputed or deleted.