You've got quite a collection of genes here.
To start with, both parents should have the genes to be black all over. That's one major gene and several others that help. But as long as both parents are pure for all the genes involved, you can ignore that part of it.
The hen has the lavender gene. It is recessive, so a chicken must inherit it from both parents in order to show any effect. Since only the hen has that gene, not the rooster, then every chick will inherit one lavender gene and will not show any visible effects.
Lavender dilutes black to a light gray shade, and dilutes red/gold shades to light yellow (if your chickens have no red/gold/yellow, that part will not matter.)
The Mauve rooster has one blue gene and two chocolate genes.
Blue is an incompletely dominant gene. One blue gene turns black into blue (a gray color), or it turns chocolate into mauve. Two blue genes turn black into splash (a very light gray, usually with black blotches in places). I don't know what to call a chicken with splash and chocolate, and I don't know what it looks like, but you won't get any from this breeding anyway.
When you breed a blue chicken to a chicken with no blue gene, the not-blue parent gives every chick a not-blue gene. The blue parent gives a blue gene to half the chicks (so they look blue) and a not-blue gene to the other half of the chicks (so they look black, unless there are other genes affecting their color.)
Blue and splash affect black but not red/gold colors (that is why there can be Blue Laced Red Wyandottes, where the black turns to blue but the red stays red, or Splash Laced Red Wyandottes where the black turns to splash but the red still stays red.) If your chickens have no red/gold/yellow colors, that will not matter. But if you are ever trying to sort blue from lavender in chickens that have other colors as well, you can sometimes tell by looking for red or gold (blues) or pale yellow (probably lavenders.)
The barring gene and the chocolate gene are both on the Z sex chromosomes. Barring is dominant and chocolate is recessive.
Roosters have sex chromosomes ZZ. They inherit one Z from each parent, and give one Z to each chick.
Your mauve rooster must have chocolate on both of his Z chromosomes, because chocolate is recessive (so if he only had chocolate on one Z chromosome, he would not look mauve.)
Hens have sex chromosomes ZW. They inherit Z from their father and give it to their sons. They inherit W from their mother and give it to their daughters (so the hen determines which chicks are male or female: backwards of humans and many other mammals.)
Your cuckoo hen has barring on her Z chromosome. Barring is dominant, but that doesn't matter in a hen, because she only has one Z chromosome. She shows whatever is on that Z chromosome. She will give a Z chromosome (with barring) to her sons, and a W chromosome to her daughters.
From this pair, the sons inherit a Z chromsome with chocolate from their father, and a Z chromosome with barring from their mother. Because barring is dominant, the sons show barring. Because chocolate is recessive, the sons do not show chocolate (the dominant not-chocolate gene on the mother's Z chromosome will make her sons show not-chocolate.)
The daughters inherit a Z chromosome with chocolate from their father, and a W chromosome from their mother. The W chromosome makes them female, but doesn't have any other effect that we care about here. The Z chromosome has chocolate, so the daughters show chocolate because they do not have a dominant not-chocolate gene to hide it.
Taking all of that together is what gives the chick sexs & colors I predicted in my first response. I think I accounted for lavender first (all chicks will carry it, none will show it), then I figured out the sex-linked genes (barring and chocolate), then I accounted for half of the chicks getting blue and half not. Then I double-checked myself a time or two, because it's easy to forget something when dealing with that many genes at once
