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Math 3533 Spring 2020 FE-TH. Prof. Iarrobino Name .............................
Please show your work and explain your reasoning for full credit
Understanding for Take Home part of FE: Open textbook; you may consult freely posted materials
at Math 3533 Blackboard, including Daily Progress, and Project Reports and R. Stanleys book
on Algebraic Combinatorics (the six chapters posted); you may work with another current student
of Math 3533 on a problem, provided you understand what you write and report this (who you
worked with). I would encourage students not to use online sources outside of the Blackboard
provided, but if you do use, you must report which and give references. You are welcome to ask
me questions, but I prefer you not ask others outside of the class, without consulting me first: any
such discussions are to be reported.
Due Tuesday April 21 at 10 AM, please submit using Blackboard.
1A. How many ways are there to choose a bouquet of n flowers from daffodils, tulips, and
daises, provided there must be at least two daffodils?
P
1B. Write an ordinary generating function B =
bn xn , where bn counts the number of
different ways of choosing n fruit for a fruit basket comprised of apples, oranges, pears
and kumquats, provided that the apples come in packets of 3, there are an even number
of pears, and an odd number at least 3 of kumquats. We regard the fruit of each kind as
identical.
P
1C.i. Give an exponential generating function E(x) =
en xn /n! where en counts words
made of the letters A, B, C, D, with an even number of A’s, and odd number of B’s, at
least one C, and at most two D’s. Your generating function should be the product of four
functions, one for each of the letters.
ii. Use your work to obtain a closed expression for en , in terms of powers and binomial
coefficients involving n.
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Math 3533 Spring 2020, FE, TH
2A. Six nonattacking rooks (they must be in different rows and columns) are placed
on a 8 × 8 chess board, and are colored from a palette of three colors red, green, and blue.
How many different placements of colored rooks are there? (The board does not rotate.)
2B. How many directed paths are there from the lower left corner (0, 0) to the upper
right corner (9, 9) of a 9 × 9 board?
2C. How many of those paths do not pass through either (3, 2) or (6, 4)?
2D*. How many of those paths do not go above the diagonal {(k, k) | 1 ≤ k ≤ 8 (that
is, each vertex (a, b) on the path satisfies a ≥ b)?
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Math 3533 Spring 2020 FE-TH.
3A. Consider the poset P = V × [n] where the poset V is the claw poset with 3 vertices
{a, b, c | a < b, a < c}, and the poset [n] is the string 1 < 2 < · · · < n. Recall (March
P i
9 Daily Progress but changed notation) that the height polynomial polyq (P ) =
ri q
where ri is the number of vertices at height i from the bottom: so polyq (V ) = 1 + 2q, and
Polyq ([n]) = 1 + q + · · · + q n−1 . The height polynomial is multiplicative.
3Ai. Determine the number of vertices and edges (between adjacent vertices) of P , in
terms of n. Draw or describe carefully a Hasse diagram for P .
3Aii. Determine the chain partitions C(P ), C(V ) and C([n]). For C(P ) determine a
single chain having the maximum number of vertices, and two chains having a maximum
number of vertices.
Also determine the antichain partition A(P ) for P , and specify its relation with C(P )
and with polyq (P ).
3Aiii. A poset P is Sperner if the maximum magnitude of any antichain is the maximum coefficient of polyq (P ). Is P Sperner? (The Sperner property is discussed in Ty’s
presentation).
Can you find a condition on a poset Q such that V × Q is Sperner, V the claw poset?
3Aiv. Consider P2 = V × [2], which has 6 vertices and 7 edges. A linear extension of
the partial order is to view the vertices in the order (a, 1), (b, 1), (a, 2), (b, 2), (c, 1), (c, 2).
Recall that ζi,j = 1 if i ≤ j. Determine the 6 × 6 matrix ζ − I for P2 and the Mobius
matrix µ(P2 )?
3Av. Also determine the non-zero entries of (ζ − I)3 .
3Avi. Prove that Bn (the set of subset of {1, 2, . . . , n} has the Sperner property. (Hint,
one way involves using properties of products of certain posets).
3B*. Analogous questions to those of 3Ai-iii for the poset W = D × V where the poset
D = Div(72) of the 12 divisors of 72 = 23 · 32 , partially ordered by division, and V is again
the claw poset.
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Figure 1. Flow diagram. (by Ian Quain, Nick Ratliff.)
4. (Related to project of Ian and Nick R.).
4A. Determine the maximum flow (and the routing) for the flow diagram above. Show
two steps in a method for finding it.
4B. Explain the Ford-Fulkerson max-flow/minimum cut theorem as it applies to this
diagram.
5
Math 3533 Spring 2020. FE TH part.
5. (Related to the project of Tinglei and He- see also R. Stanley Algebraic Combinatorics, Chap 6, posted and Daily Progress.)
P
n
The quantum binomial coefficient satisfies
= λ⊂(n−k)k q |λ| , that is, the coeffik q
cient of q s counts the number of partitions of s whose Ferrers diagram fits in a k × (n − k)
box (the box has k rows of length n − k).
5A. Fixing k and n, determine the total number of partitions p(k, n) whose Ferrers
diagram fits into the k × (n − k) box in two ways.
i. Compare these partitions with the number of Northeast paths in a lattice from (0, 0)
to (n − k, k). Explain the comparison.
Give a closed expression for p(k, n).
n
ii. Evaluate the limitq→1
. Illustrate your limit in the case n = 5, k = 2.
k q
5B. For the case n = 5, k = 3 determine the number of partitions of each of 7, , 8 and
of 9 whose Ferrers shape fits in to the box, in two ways.
5C. Consider the poset P(k, n) of partitions that fit into the k × n box B, under the
partial order induced by inclusion of the poset diagrams. Prove that the map τ : P →
P c = B\C taking P into its complement in B is an involution on P(k, n). You must show
that τ is a bijection (1-1, onto), and that P < Q ⇒ P c ⊂ Qc .
5D. Consider the poset P (k, k). Show that the map ι : P → P ∨ is an involution: that
is, show ι2 =identity, and if P < Q then ι(P ) < ι(Q).
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Math 3533 Spring 2020 FE TH.
6A. Determine the number of permutations in S16 having cycle partition (34 , 22 ) (indicate answer and reasoning).
6B. Which Stirling number counts all the partitions of the set [9] = {1, 2, . . . 9} into 3
distinct subsets?
6C. Determine the number of surjective maps from the set [9] to the set [3] = {1, 2, 3}.
(You may use the Stirling number formulas to indicate the answer as a sum.) Explain the
relation of thie problem to # 6B, and also explain how to determine the formula
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