A Tiling and (0, 1)-Matrix Existence Problem

Transcription

1 A Tiling and (0, )-Matrix Existence Problem Curtis G Nelson University of Wyoming Advisor: Dr. Bryan Shader July 24, 205

2 A (0, )-Matrix and Tiling Problem Let R = (r, r 2,..., r m, r m+ = 0) and S = (s, s 2,..., s n ) be nonnegative integral vectors.

3 A (0, )-Matrix and Tiling Problem Let R = (r, r 2,..., r m, r m+ = 0) and S = (s, s 2,..., s n ) be nonnegative integral vectors. Question: Can a (m + ) n checkerboard be tiled with vertical dimers and monomers so that there are r i dimers with the upper half of the dimer in row i and s i dimers in column i?

4 A (0, )-Matrix and Tiling Problem Example: R = (2, 2,, 2, 0); S = (2,, 2, 2)

5 A (0, )-Matrix and Tiling Problem Example: R = (2, 2,, 2, 0); S = (2,, 2, 2)

6 A (0, )-Matrix and Tiling Problem Example: R = (2, 2,, 2, 0); S = (2,, 2, 2) 0

7 A (0, )-Matrix and Tiling Problem Example: R = (2, 2,, 2, 0); S = (2,, 2, 2)

8 Other Ways to Phrase the Question. A question about the existence of a (0, )-matrix where every sequence of s in a column has an even number of s.

9 Other Ways to Phrase the Question. A question about the existence of a (0, )-matrix where every sequence of s in a column has an even number of s. 2. The existence of a (0, )-matrix where no consecutive s occur in a column.

10 Other Ways to Phrase the Question. A question about the existence of a (0, )-matrix where every sequence of s in a column has an even number of s. 2. The existence of a (0, )-matrix where no consecutive s occur in a column. 3. Phrase it as a linear programming problem and look for a 0, solution. (a + a a n = r, etc.)

11 Our Point of View The existence of a (0, )-matrix where no consecutive s occur in a column.

12 Definition Definition Let A(R, S) be the set of all (0, )-matrices with row sum vector R column sum vector S.

13 Definition Definition Let A(R, S) be the set of all (0, )-matrices with row sum vector R column sum vector S. -Studied by H.J. Ryser, D. Gale, D.R. Fulkerson, R.M Haber, and R. Brualdi.

14 Definition Definition Let A (R, S) be the set of all (0, )-matrices with row sum vector R column sum vector S no consecutive s occur in any column.

15 Definition Definition Let A (R, S) be the set of all (0, )-matrices with row sum vector R column sum vector S no consecutive s occur in any column. Question When is A (R, S) nonempty?

16 Example R = (,, 3, 2, 2, 3); S = (3,, 3,,, 3)

17 Example R = (,, 3, 2, 2, 3); S = (3,, 3,,, 3)

18 Example R = (,, 3, 2, 2, 3); S = (3,, 3,,, 3)

19 Example R = (,, 3, 2, 2, 3); S = (3,, 3,,, 3) X X

20 An Observation Observation: If M A (R, S) then we can entry wise sum rows r i and r i+ and get a matrix in A((r,..., r i, r i + r i+, r i+2,... ), S).

21 An Observation Observation: If M A (R, S) then we can entry wise sum rows r i and r i+ and get a matrix in A((r,..., r i, r i + r i+, r i+2,... ), S). The Gale-Ryser Theorem characterizes when A(R, S) is nonempty.

22 Definition Majorization: Nonincreasing integral vectors: a = (a, a 2,... a n ) and b = (b, b 2,...b m ). -append zeros to make them equal length (say n m). a is majorized by b, denoted a b when and a + a a k b + b b k for all k a + a a n = b + b b n

23 Definition Majorization: Nonincreasing integral vectors: a = (a, a 2,... a n ) and b = (b, b 2,...b m ). -append zeros to make them equal length (say n m). a is majorized by b, denoted a b when and a + a a k b + b b k for all k a + a a n = b + b b n

24 Example Example: (3, 2,, 0) (3, 3, 0, 0) since = =

25 Definition Conjugate of a nonnegative integral vector: R = (3, 2, 3, ) R = (4, 3, 2, 0,..., 0)

26 Definition Conjugate of a nonnegative integral vector: R = (3, 2, 3, ) R = (4, 3, 2, 0,..., 0)

27 Existence Theorem for A(R, S) Theorem (Gale-Ryser, 957) If R = (r, r 2,..., r m ) and S = (s, s 2,..., s n ) are nonnegative integral vectors such that S is nonincreasing, then there exists an m n, (0, )-matrix with row sum vector R and column sum vector S if and only if S R.

28 Existence Theorem for A(R, S) Theorem (Gale-Ryser, 957) If R = (r, r 2,..., r m ) and S = (s, s 2,..., s n ) are nonnegative integral vectors such that S is nonincreasing, then there exists an m n, (0, )-matrix with row sum vector R and column sum vector S if and only if S R. For A (R, S), S (r + r 2, r 3 + r 4, r 5 ) S (r + r 2, r 3, r 4 + r 5 ) S (r, r 2 + r 3, r 4 + r 5 )

29 Definition Let Q R = {(r + r 2, r 3 + r 4, r 5 ), (r + r 2, r 3, r 4 + r 5 ), (r, r 2 + r 3, r 4 + r 5 )}.

30 Definition Let Q R = {(r + r 2, r 3 + r 4, r 5 ), (r + r 2, r 3, r 4 + r 5 ), (r, r 2 + r 3, r 4 + r 5 )}. Observation: If A (R, S) is nonempty then S q for all q Q R.

31 Definition Let Q R = {(r + r 2, r 3 + r 4, r 5 ), (r + r 2, r 3, r 4 + r 5 ), (r, r 2 + r 3, r 4 + r 5 )}. Observation: If A (R, S) is nonempty then S q for all q Q R. Is this condition sufficient to show A (R, S) is nonempty?

32 Existence Theorem for A (R, S) Theorem (N., Shader) If R = (r, r 2,..., r m ) and S = (s, s 2,..., s n ) are nonnegative integral vectors such that S is nonincreasing, then there exists an m n (0, )-matrix with no two s occurring consecutively in a column and with row sum vector R and column sum vector S if and only if S q q Q R.

33 Proofs of the Gale-Ryser Theorem -direct combinatorial arguments -network flows

34 Proofs of the Gale-Ryser Theorem -direct combinatorial arguments -network flows S r r 2 r 3 r 4 r 5 x x 2 x 3 x 4 x 5 y y 2 y 3 y 4 y 5 s s 2 s 3 s 4 s 5 T

35 Network Flow for A(R, S) S 2 x x 2 0 y y 2 2 T x 3 y

36 For A(R, S) (0, ) matrix Flow Conditions

37 5 q r r 2 + r 3 r 4 + r 5 x x 2 x 3 y y 2 y 3 s s 2 s 3 t 5 i= r i i= r i S q 2 T q 3

38 Existence Theorem for A (R, S) Proof. Main idea: Induction

39 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns).

40 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns). Induct on s n (the number of s in the last column).

41 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns). Induct on s n (the number of s in the last column). Carefully choose a r i to decrease by and decrease s n by.

42 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns). Induct on s n (the number of s in the last column). Carefully choose a r i to decrease by and decrease s n by. Argue that the inductive hypotheses hold.

43 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns). Induct on s n (the number of s in the last column). Carefully choose a r i to decrease by and decrease s n by. Argue that the inductive hypotheses hold. Let M be a (0, )-matrix in A ((r, r 2,..., r i,..., r m), (s,..., s n )).

44 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns). Induct on s n (the number of s in the last column). Carefully choose a r i to decrease by and decrease s n by. Argue that the inductive hypotheses hold. Let M be a (0, )-matrix in A ((r, r 2,..., r i,..., r m), (s,..., s n )). If there is a in the (i, n) position, argue that with a switch this can be changed to a 0.

45 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns). Induct on s n (the number of s in the last column). Carefully choose a r i to decrease by and decrease s n by. Argue that the inductive hypotheses hold. Let M be a (0, )-matrix in A ((r, r 2,..., r i,..., r m), (s,..., s n )). If there is a in the (i, n) position, argue that with a switch this can be changed to a 0. Put a in the (i, n) position and use switches to remove any consecutive s.

46 Existence Theorem for A (R, S) Proof. Main idea: Induction Induct on n (the number of columns). Induct on s n (the number of s in the last column). Carefully choose a r i to decrease by and decrease s n by. Argue that the inductive hypotheses hold. Let M be a (0, )-matrix in A ((r, r 2,..., r i,..., r m), (s,..., s n )). If there is a in the (i, n) position, argue that with a switch this can be changed to a 0. Put a in the (i, n) position and use switches to remove any consecutive s. This completes the induction on the number of s in the last column and in turn the induction on the number of columns.

47 Graph of A (R, S) Definition The graph of A (R, S) is an undirected graph with: vertices are the matrices in A (R, S) M M 2 if and only if the matrix M can be changed to M 2 with one basic switch.

48 Further Work Let M A (R, S). What is the probability that a occurs in position m ij?

49 Further Work Let M A (R, S). What is the probability that a occurs in position m ij? Determine statistical information about A (R, S) by studying a Markov chain defined on the graph of A (R, S).

50 Further Work Let M A (R, S). What is the probability that a occurs in position m ij? Determine statistical information about A (R, S) by studying a Markov chain defined on the graph of A (R, S). What if every is followed by j zeros: A j (R, S)?

51 Further Work Let M A (R, S). What is the probability that a occurs in position m ij? Determine statistical information about A (R, S) by studying a Markov chain defined on the graph of A (R, S). What if every is followed by j zeros: A j (R, S)? Further study of the network flow connection.

52 Further Work (0, ) matrix A(R, S) : Flow (0, ) matrix Conditions A (R, S) : Flow Conditions

53 References [] R. Brualdi Combinatorial Matrix Classes. Cambridge University Press, New York, [2] R. Brualdi and H. Ryser Combinatorial Matrix Theory. Cambridge University Press, New York, 99. [3] L. Ford and D. Fulkerson Maximal flow through a network. Canadian Journal of Mathematics, 8:399, 956. [4] J. van Lint and R. Wilson A Course in Combinatorics. Cambridge University Press, New York, 992.