Chained Permutations. Dylan Heuer. North Dakota State University. July 26, 2018

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1 Chained Permutations Dylan Heuer North Dakota State University July 26, 2018

2 Three person chessboard

3 Three person chessboard

4 Three person chessboard

5 Three person chessboard - Rearranged

6 Two new families of chessboards The board B 5,3 The board B 4,6

7 General enumerative result Theorem The number of ways to place m non-attacking rooks on board B {B n,k, B n,k } is (a 1,...,a k ) C m (B) i=1 k ( ) n ai 1 (n) ai where a 0 is defined as: { 0 if B = B n,k a 0 = a k if B = Bn,k. a i

8 Chained permutations Maximum rook placement: Permutation matrix form:

9 Chained permutations Permutation matrix form: One-line notation:

10 With usual permutations, we can use adjacent transpositions to obtain weak order.

11 With usual permutations, we can use adjacent transpositions to obtain weak order. Thinking of a permutation in matrix form, we can think of an adjacent transposition as swapping adjacent rows.

12 With usual permutations, we can use adjacent transpositions to obtain weak order. Thinking of a permutation in matrix form, we can think of an adjacent transposition as swapping adjacent rows. There is a natural way to modify this in the case of chained permutations.

13 With usual permutations, we can use adjacent transpositions to obtain weak order. Thinking of a permutation in matrix form, we can think of an adjacent transposition as swapping adjacent rows. There is a natural way to modify this in the case of chained permutations. We can perform a swap of adjacent rows on the ith matrix, while simultaneously performing a corresponding swap of adjacent columns on the (i + 1)st matrix.

14 s 3,

15 We can use these transpositions to generate a poset, just like with usual permutations.

16 We can use these transpositions to generate a poset, just like with usual permutations. SageMath has been useful not only for its computational power, but also for its ability to visualize and work with graphs and posets.

17 k = 2, n = 3, circular, fixed composition (2,1)

18 Inversion number? It seems that there is a relatively nice analog of inversion number for chained permutations.

19 s 3,

20 s 1,

21 s 2,

22 s 3,

23 Inversion number? It seems that there is a relatively nice analog of inversion number for chained permutations.

24 Inversion number? It seems that there is a relatively nice analog of inversion number for chained permutations. We can start with a chained permutation and algorithmically change it to the identity.

25 Inversion number? It seems that there is a relatively nice analog of inversion number for chained permutations. We can start with a chained permutation and algorithmically change it to the identity. In fact, it appears to be the case that using this analog of inversion number, k [ ] n q inv(w) ai 1 = [n] ai a i w P n,k i=1 q (the q-analog of the counting formula), just as it is with usual permutations.

26 Thank you!

1111: Linear Algebra I

1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 7 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 7 1 / 8 Invertible matrices Theorem. 1. An elementary matrix is invertible. 2.