# Algebraic Combinatorics by Ulrich Dempwolff

By Ulrich Dempwolff

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Extra info for Algebraic Combinatorics

Example text

R ∈ [k]. X m we get k αr = Xk 1 − rX r=1 k αr r m X m = m≥0 41 ( m≥0 r=1 αr rm )X m+k . (k − r)! (k − r)! 5. Definition (a) Let A be a finite set called an alphabet and W(A) = {w∅ } An . n≥1 The set W(A) is called the set of words over A. The symbol w∅ is called the empty word. One writes w = a1 . . an for a word w ∈ An instead of w = (a1 , . . , an ). In this case |w| = n is the length of w. One defines |w∅ | = 0. (b) Let w ∈ W(A) of length k ≥ 1. Define Ww (n) be the set of words v of length n such that v does not contain w as a subword.

K + 1). Then (λ∗ )∗ = λ and (µ∗ )∗ = µ which shows |Λ2 | = pk (n − k). ✷ With the help of this formula we compute the first values of the p(n)’s. n p(n) 1 1 2 2 3 3 4 5 5 7 6 11 7 15 8 22 9 30 10 42 ... Definition Let λ = (λ1 , . . , λk ) be a partition of n. Reverse the order of the y-axis in the coordinate system of the real plane. The collection of the lattice points with the coordinates (1, 1) (2, 1) · · · (1, 2) (2, 2) · · · ··· ··· ··· (1, k) (2, k) · · · (λ1 − 2, 1) (λ1 − 1, 1) (λ1 , 1) (λ1 − 1, 2) (λ1 , 2) ··· (λk − 1, k) is the Ferres diagram.

Replacing the dots by square boxes we obtain the Young diagram which is a useful notion in the representation theory of the symmetric groups. Partition λ =(5,3,1,1) Define the partition λ = (λ1 , . . , λλ1 ) by λj = |{i | λi ≥ j}|. Then λ is the conjugate partition of λ. The Ferres-diagram of λ is obtained by reflecting the Ferres-diagram of λ on the line with slope −1 through the origin. One has λ=λ . 2 pk (n) is also the number of partitions of n with a largest part of size k. Proof. Apply conjugation.