Combinatorics

By Martin Aigner

Combinatorial enumeration is a with ease available topic packed with simply acknowledged, yet occasionally tantalizingly tough difficulties. This ebook leads the reader in a leisurely method from the fundamental notions to numerous subject matters, starting from algebra to statistical physics. Its goal is to introduce the coed to a fascinating box, and to be a resource of data for the pro mathematician who desires to examine extra concerning the topic. The booklet is equipped in 3 elements: fundamentals, tools, and subject matters. There are 666 workouts, and as a distinct characteristic each bankruptcy ends with a spotlight, discussing a very appealing or recognized result.

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Extra info for A Course in Enumeration

Sample text

The following ﬁgure demonstrates a natural bijection between these paths and the set Par( ; ≤ n; ≤ m), by interpreting the part above the lattice path as a Ferrers diagram. Example. (5, 4) n=4 (0, 0) → λ = 4311 m=5 Hence p( ; ≤ n; ≤ m) = m+n . m (5) Note that the path that goes up to y = n and then horizontally to (m, n) corresponds to the empty partition. In our example above, n = 3, m = 2, we obtain p( ; ≤ 3; ≤ 2) = 2+3 = 10. 2 Summary. An easy way to remember some of the fundamental coeﬃcients we have encountered so far is to interpret them as distributions of n balls into r boxes.

2. 3. 4. S(T ) is a partial tiling, S(T ) has no black blocks, S S(T ) = T , S(T ) is reduced. To prove (1) suppose to the contrary that a white cell s is covered twice in S(T ) (the case of a black cell is analogous). Then we have in T without loss of generality the following situation: s or s Since T contains no black blocks, s is a free cell of T . But then s could not be in a free black block, which is impossible, since T is reduced. 48 1 Fundamental Coeﬃcients The assertion (2) is clear, since black blocks are left invariant under S, and (3) is directly implied by the deﬁnition of shuﬄing.

K} (the xi ’s being the multiplicities). Hence n+k−1 their number is . n Partition Numbers. Back to ordinary unordered partitions. We have introduced the notation Par(n), Par(n; k). Similarly, we use Par(n; ≤ k) for the set 32 1 Fundamental Coeﬃcients of all partitions of n with at most k parts, and set p(n; ≤ k) = |Par(n; ≤ k)|. For the partition numbers p(n; k) there is no recurrence of order 2. Instead, we have p(n; k) = p(n − k; ≤ k) = p(n − k; 1) + · · · + p(n − k; k) . (1) Indeed, the mapping φ : Par(n; k) → Par(n−k; ≤ k) with φ(λ1 .