#You may post this HW.
#Richard Voepel, Math 640 (Fall 2016) Homework Assignment 3

with(ListTools):


Help:=proc(): print(`WtE(m,S,t): Generating function for compositions with components in specified residue classes given by S and m.`): end:


#1) I am skipping this problem due to my familiarity with Maple.


#2) My apologies, but I did not get around to actually writing up my one page summary. It is a very good overview of the first half of Algebraic Combinatorics, the abstraction of the concrete as you have put it, A number of the techniques discussed in the paper I had not ever really worked with or heard of before, but I think that the majority are at least famous or common enough that I had some exposure to them (though I do not claim expertise in any). Presumably we will become intimately familiar with all of them by the end of the semester.


#3) Below you will find WtE(m,S,t), a procedure that outputs a generating function in t for the number of compositions with prescribed components based on the residue classes of the elements of S mod m. In these cases, the output will be a rational function, and the coefficients of the Maclaurin expansion of that function will be the desired sequence.

# WtE(5,{1,4},t) = (t^5-1)/(t^5+t^4+t-1)
# WtE(7,{3,7},t) = (t^7-1)/(2*t^7+t^3-1)

WtE:=proc(m,S,t) local s,A,a:

A:={}:
for s in S do
  A:=A union {s mod m}:
od:

if evalb(0 in A) then
  A:=A union {m}:
  A:=A minus {0}:
fi:

RETURN(simplify((1-add(t^a/(1-t^m),a in A))^(-1))):

end:


#4) I did not attempt the Human Challenge due to time constraints.