SLIDE 1 Alejandro Morales Joel B. Lewis (Minnesota) Counting matrices over finite fields with zeroes
CanaDAM 2013, Minisymposia enumerative combinatorics June 10, 2013 joint work with Aaron Klein (Brookline high school ! MIT)
R
(LaCIM, Universit´ e du Qu´ ebec ` a Montr´ eal)
SLIDE 2
q = 1: rook placements
Sn = {permutations w of {1, 2, . . . , n}} Let S ✓ {1, 2, . . . , n} ⇥ {1, 2, . . . , n}
SLIDE 3
q = 1: rook placements
Sn = {permutations w of {1, 2, . . . , n}} Let S ✓ {1, 2, . . . , n} ⇥ {1, 2, . . . , n} p(n, S) = #{permutations w = w1 · · · wn | wi 6= j if (i, j) 2 S} = #{placements of n non-attacking rooks on S} = #{n ⇥ n permutation matrices P | Pi,j = 0 if (i, j) 2 S}
SLIDE 4 q = 1: rook placements
Sn = {permutations w of {1, 2, . . . , n}} Let S ✓ {1, 2, . . . , n} ⇥ {1, 2, . . . , n} p(n, S) = #{permutations w = w1 · · · wn | wi 6= j if (i, j) 2 S} = #{placements of n non-attacking rooks on S} = #{n ⇥ n permutation matrices P | Pi,j = 0 if (i, j) 2 S}
w = 4312
0 0 0 R R R R 0 0 1 1 1 1
SLIDE 5 q = 1: rook placements
Sn = {permutations w of {1, 2, . . . , n}} Let S ✓ {1, 2, . . . , n} ⇥ {1, 2, . . . , n} Examples
Ferrers board: Diagram of w:
0 0 0
S = Dw S = Fλ w = 2143
p(n, S) = #{permutations w = w1 · · · wn | wi 6= j if (i, j) 2 S} = #{placements of n non-attacking rooks on S} = #{n ⇥ n permutation matrices P | Pi,j = 0 if (i, j) 2 S}
λ = (1, 2, 4, 4) λ = (1, 2, 4, 4)
SLIDE 6
q: matrices over Fq with restricted support
GL(n, q) = {n ⇥ n invertible matrices over Fq} Fq finite field q = ps elements Let S ✓ {1, 2, . . . , n} ⇥ {1, 2, . . . , n} mq(n, S) = #{matrices A 2 GL(n, q) | Aij = 0 if (i, j) 2 S} 1 2 2 1 q = 3 1 1 1 1 2 2
SLIDE 7 q: matrices over Fq with restricted support
GL(n, q) = {n ⇥ n invertible matrices over Fq} Fq finite field q = ps elements Let S ✓ {1, 2, . . . , n} ⇥ {1, 2, . . . , n} mq(n, S) = #{matrices A 2 GL(n, q) | Aij = 0 if (i, j) 2 S} Examples
Ferrers board: Diagram of w:
0 0 0
S = Dw S = Fλ w = 2143 λ = (1, 2, 4, 4)
1 2 2 1 q = 3 1 1 1 1 2 2
SLIDE 8
q: matrices over Fq with restricted support
SLIDE 9 Examples p(n, S) and mq(n, S)
p(n, S) = #{placements of n non-attacking rooks on S} 0 0 0
1 choices 2 1 ” 4 2 ” 4 3 ” p(n, Fλ) =
n
Y
i=1
(λi i + 1)
S = Fλ
λ = (1, 2, 4, 4) total 2 choices
SLIDE 10 Examples p(n, S) and mq(n, S)
p(n, S) = #{placements of n non-attacking rooks on S} 0 0 0
1 choices 2 1 ” 4 2 ” 4 3 ” p(n, Fλ) =
n
Y
i=1
(λi i + 1)
S = Fλ
λ = (1, 2, 4, 4) total 2 choices
mq(n, S) = #{matrices A 2 GL(n, q) | Aij = 0 if (i, j) 2 S} 0 0 0
q1 1 choices q2 q1 ” q4 q2 ” q4 q3 ” λ = (1, 2, 4, 4)
SLIDE 11 Examples p(n, S) and mq(n, S)
p(n, S) = #{placements of n non-attacking rooks on S} 0 0 0
1 choices 2 1 ” 4 2 ” 4 3 ” p(n, Fλ) =
n
Y
i=1
(λi i + 1)
S = Fλ
λ = (1, 2, 4, 4) total 2 choices
mq(n, S) = #{matrices A 2 GL(n, q) | Aij = 0 if (i, j) 2 S} 0 0 0
q1 1 choices q2 q1 ” q4 q2 ” q4 q3 ” mq(n, Fλ) =
n
Y
i=1
qλi qi1 λ = (1, 2, 4, 4)
SLIDE 12 Examples p(n, S) and mq(n, S)
p(n, S) = #{placements of n non-attacking rooks on S} 0 0 0
1 choices 2 1 ” 4 2 ” 4 3 ” p(n, Fλ) =
n
Y
i=1
(λi i + 1)
S = Fλ
λ = (1, 2, 4, 4) total 2 choices
mq(n, S) = #{matrices A 2 GL(n, q) | Aij = 0 if (i, j) 2 S} 0 0 0
q1 1 choices q2 q1 ” q4 q2 ” q4 q3 ” mq(n, Fλ) =
n
Y
i=1
qλi qi1 = (q 1)nq(
n 2)
n
Y
i=1
[λi i + 1]q where [k]q = 1 + q + · · · + qk1 λ = (1, 2, 4, 4)
SLIDE 13 Remarks and outline
Remarks
- p(n, S), mq(n, S) invariant under permuting rows and
columns of S
SLIDE 14 Remarks and outline
Remarks
- p(n, S), mq(n, S) invariant under permuting rows and
columns of S i) mq(n, Fλ)/(q 1)n is in N[q],
SLIDE 15 Remarks and outline
Remarks
- p(n, S), mq(n, S) invariant under permuting rows and
columns of S ,
an interval in the Bruhat order
i) mq(n, Fλ)/(q 1)n is in N[q], ii) p(n, F λ) = # ( )
SLIDE 16 Remarks and outline
I Is mq(n, S)/(q 1)n a polynomial in q? Is it in N[q]? II When are rook placements on S related to Bruhat intervals? Remarks
- p(n, S), mq(n, S) invariant under permuting rows and
columns of S , Outline
an interval in the Bruhat order
i) mq(n, Fλ)/(q 1)n is in N[q], ii) p(n, F λ) = # ( )
SLIDE 17 Is mq(n, S)/(q 1)n a polynomial in N[q]?
If S = {(1, 1), (2, 2), (3, 3)}, mq(3, S) = (q 1)3(q3 + 2q2 q)
Example
SLIDE 18 Is mq(n, S)/(q 1)n a polynomial?
1 3 6 7 2 5 4 SP G(2,2) Fano plane PG(2, 2)
(Stembridge 1998) Example
SLIDE 19 Is mq(n, S)/(q 1)n a polynomial?
mq(7, SP G(2,2)) = ⇢
(q − 1)7(q14 + · · · −97q9 + · · · +q3) if q even, (q − 1)7(q14 + · · · −98q9 + · · · −6q5) if q odd.
1 3 6 7 2 5 4 SP G(2,2) Fano plane PG(2, 2)
(Stembridge 1998) Example
SLIDE 20 however, mq(n, S) is a q-analogue of p(n, S)
Theorem (Lewis, Liu, M, Panova, Sam, Zhang 2011) For all S ⇢ {1, . . . , n} ⇥ {1, . . . , n}, mq(n, S) ⌘ p(n, S) · (q 1)n (mod (q 1)n+1).
– If S = {(1, 1), (2, 2), (3, 3)}, mq(3, S)/(q 1)3 = q3 + 2q2 q mq(3, S)/(q 1)3
SLIDE 21 however, mq(n, S) is a q-analogue of p(n, S)
Theorem (Lewis, Liu, M, Panova, Sam, Zhang 2011) For all S ⇢ {1, . . . , n} ⇥ {1, . . . , n}, mq(n, S) ⌘ p(n, S) · (q 1)n (mod (q 1)n+1). Examples
– If S = {(1, 1), (2, 2), (3, 3)}, mq(3, S)/(q 1)3 = q3 + 2q2 q mq(3, S)/(q 1)3
SLIDE 22 however, mq(n, S) is a q-analogue of p(n, S)
Theorem (Lewis, Liu, M, Panova, Sam, Zhang 2011) For all S ⇢ {1, . . . , n} ⇥ {1, . . . , n}, mq(n, S) ⌘ p(n, S) · (q 1)n (mod (q 1)n+1). Examples
– If S = {(1, 1), (2, 2), (3, 3)}, mq(3, S)/(q 1)3 = q3 + 2q2 q mq(3, S)/(q 1)3
– mq(7, SP G(2,2))/(q 1)7
⇢
24 if q even, 24 if q odd.
SLIDE 23
Outline of talk
⇥
I Is mq(n, S)/(q 1)n a polynomial in q?, Is it in N[q]? For which S is it in N[q]? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw? II When are rook placements on S related to Bruhat intervals?
SLIDE 24 mq(n, Fλ)/(q 1)n is in N[q]
Theorem (Haglund 1998) mq(n, Fλ) = (q 1)n X
w2p(n,Fλ)
qboxNE(w).
boxes in Fλ strictly North and East of rooks
SLIDE 25 mq(n, Fλ)/(q 1)n is in N[q]
Theorem (Haglund 1998) mq(n, Fλ) = (q 1)n X
w2p(n,Fλ)
qboxNE(w). Example
boxes in Fλ strictly North and East of rooks λ = (1, 2, 4, 4)
SLIDE 26 mq(n, Fλ)/(q 1)n is in N[q]
Theorem (Haglund 1998) mq(n, Fλ) = (q 1)n X
w2p(n,Fλ)
qboxNE(w). Example
boxes in Fλ strictly North and East of rooks mq(4, F1244) = (q 1)4
+ q7
λ = (1, 2, 4, 4)
SLIDE 27 mq(n, Fλ)/(q 1)n is in N[q]
Theorem (Haglund 1998) mq(n, Fλ) = (q 1)n X
w2p(n,Fλ)
qboxNE(w). Example
boxes in Fλ strictly North and East of rooks mq(4, F1244) = (q 1)4
+ q7 Haglund’s result extends to skew Ferrers shapes S = Fλ/µ Proposition (Klein-Lewis-M 2012)
λ = (1, 2, 4, 4)
SLIDE 28 mq(n, Fλ)/(q 1)n is in N[q]
Theorem (Haglund 1998) mq(n, Fλ) = (q 1)n X
w2p(n,Fλ)
qboxNE(w). Example
mq(4, F1244/12) = (q 1)4
+ q4 Haglund’s result extends to skew Ferrers shapes S = Fλ/µ Proposition (Klein-Lewis-M 2012)
µ = (1, 2) λ = (1, 2, 4, 4) boxes in Fλ strictly North and East of rooks
SLIDE 29
Outline of talk
⇥ X
I Is mq(n, S)/(q 1)n a polynomial in q?, Is it in N[q]? For which S is it in N[q]? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw? II When are rook placements on S related to Bruhat intervals?
SLIDE 30 mq(n, S) when S is the diagram of a permutation: Dw
Examples
w = 2431 D2431 = {(1, 1), (2, 1), (3, 1), (2, 3)} w = 2143 D2143 = {(1, 1), (3, 3)}
SLIDE 31 mq(n, S) when S is the diagram of a permutation: Dw
Examples
w = 2431 D2431 = {(1, 1), (2, 1), (3, 1), (2, 3)} w = 2143 D2143 = {(1, 1), (3, 3)}
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
SLIDE 32 mq(n, S) when S is the diagram of a permutation: Dw
Examples
w = 2431 D2431 = {(1, 1), (2, 1), (3, 1), (2, 3)} w = 2143 D2143 = {(1, 1), (3, 3)}
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q]. Can we prove the conjecture for some families of permutations?
SLIDE 33
Vexillary permutations
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
SLIDE 34 Vexillary permutations
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
- mq(n, Fλ)/(q 1)n is in N[q].
SLIDE 35 Vexillary permutations
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
- mq(n, Fλ)/(q 1)n is in N[q].
- mq(n, Dw) invariant under permuting rows/columns of Dw.
SLIDE 36 Vexillary permutations
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
- mq(n, Fλ)/(q 1)n is in N[q].
When is Dw a Ferrers shape Fλ (up to permuting rows/columns)?
- mq(n, Dw) invariant under permuting rows/columns of Dw.
SLIDE 37 Vexillary permutations
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
- mq(n, Fλ)/(q 1)n is in N[q].
When is Dw a Ferrers shape Fλ (up to permuting rows/columns)? Examples
w = 2431
- mq(n, Dw) invariant under permuting rows/columns of Dw.
SLIDE 38 Vexillary permutations
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
- mq(n, Fλ)/(q 1)n is in N[q].
When is Dw a Ferrers shape Fλ (up to permuting rows/columns)? Examples
w = 2431
0 0 ⇠
λ = (2, 3, 3, 4)
X
- mq(n, Dw) invariant under permuting rows/columns of Dw.
SLIDE 39 Vexillary permutations
Conjecture (Klein-Lewis-M 2012, true n 8) For all w 2 Sn, mq(n, Dw)/(q 1)n is in N[q].
- mq(n, Fλ)/(q 1)n is in N[q].
When is Dw a Ferrers shape Fλ (up to permuting rows/columns)? Examples
w = 2431 w = 2143
0 0 ⇠
⇥
λ = (2, 3, 3, 4)
X
- mq(n, Dw) invariant under permuting rows/columns of Dw.
SLIDE 40
Vexillary permutations
When is Dw a Ferrers shape (up to permuting rows/columns)?
SLIDE 41
Vexillary permutations
When is Dw a Ferrers shape (up to permuting rows/columns)? Dw ”is” a Ferrers shape if and only if w avoids the permutation pattern 2143. (i.e. no subsequence a < b < c < d with wb < wa < wd < wc) Theorem (Lascoux-Sch¨ utzenberger 1982)
SLIDE 42 Vexillary permutations
When is Dw a Ferrers shape (up to permuting rows/columns)?
w = 364152
Dw ”is” a Ferrers shape if and only if w avoids the permutation pattern 2143. (i.e. no subsequence a < b < c < d with wb < wa < wd < wc) Theorem (Lascoux-Sch¨ utzenberger 1982)
w = 341652
SLIDE 43 Vexillary permutations
When is Dw a Ferrers shape (up to permuting rows/columns)?
w = 364152
Dw ”is” a Ferrers shape if and only if w avoids the permutation pattern 2143. (i.e. no subsequence a < b < c < d with wb < wa < wd < wc) Theorem (Lascoux-Sch¨ utzenberger 1982)
- such permutations are called vexillary
w = 341652
SLIDE 44 Vexillary permutations
When is Dw a Ferrers shape (up to permuting rows/columns)?
w = 364152
Dw ”is” a Ferrers shape if and only if w avoids the permutation pattern 2143. (i.e. no subsequence a < b < c < d with wb < wa < wd < wc) Theorem (Lascoux-Sch¨ utzenberger 1982)
- such permutations are called vexillary
w = 341652
- there is characterization of w such that Dw is the
complement of a skew Ferrers shape (Klein-Lewis-M 2012)
SLIDE 45
Outline of talk
I Is mq(n, S)/(q 1)n a polynomial in q?, Is it in N[q]? For which S is it in N[q]? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw? ⇤ if w avoids 2143 (vexillary) II When are rook placements on S related to Bruhat intervals? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw?
⇥ X X
conjecture
X
SLIDE 46 Rook placements on Fλ and Bruhat intervals
(strong) Bruhat order () on Sn: u u · (i, j) if inv(u · (i, j)) = inv(u) + 1 321 231 213 132 123 Example (n = 3)
- [u, v] := {w | u w v} denotes an interval.
·
312
SLIDE 47 Rook placements on Fλ and Bruhat intervals
(strong) Bruhat order () on Sn: u u · (i, j) if inv(u · (i, j)) = inv(u) + 1 321 231 213 132 123 Example (n = 3)
- [u, v] := {w | u w v} denotes an interval.
Example λ = (1, 3, 3) 312 321
( ) ,
3-rook placements =
·
312
SLIDE 48 Rook placements on Fλ and Bruhat intervals
(strong) Bruhat order () on Sn: u u · (i, j) if inv(u · (i, j)) = inv(u) + 1 321 231 213 132 123 Example (n = 3)
- [u, v] := {w | u w v} denotes an interval.
Example λ = (1, 3, 3) 312 321
( ) ,
3-rook placements =
·
321 312
SLIDE 49 Rook placements on Fλ and Bruhat intervals
(strong) Bruhat order () on Sn: u u · (i, j) if inv(u · (i, j)) = inv(u) + 1 321 231 213 132 123 Example (n = 3) Theorem (Ding 2001)
- p(n, Fλ) = #[wλ, n n 1 . . . 21]
- [u, v] := {w | u w v} denotes an interval.
Example λ = (1, 3, 3) 312 321
( ) ,
3-rook placements = {n-rook placements on Fλ} = [wλ, n n 1 . . . 21] for some 213-avoiding permutation wλ.
·
321 312
SLIDE 50
Outline of talk
I Is mq(n, S)/(q 1)n a polynomial in q?, Is it in N[q]? For which S is it in N[q]? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw? ⇤ if w avoids 2143 (vexillary) II When are rook placements on S related to Bruhat intervals? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw?
⇥ X X
conjecture
X X
SLIDE 51
Rook placements on Dw and Bruhat intervals
Theorem (Lewis-M 2013) For all w, #[w, n n 1 . . . 21] p(n, Dw), = if and only if w avoids 1324, 24153, 31524, 426153.
SLIDE 52 Rook placements on Dw and Bruhat intervals
Theorem (Lewis-M 2013) For all w, #[w, n n 1 . . . 21] p(n, Dw), = if and only if w avoids 1324, 24153, 31524, 426153. Examples
321 231 312 213 132 123
w = 213
SLIDE 53 Rook placements on Dw and Bruhat intervals
Theorem (Lewis-M 2013) For all w, #[w, n n 1 . . . 21] p(n, Dw), = if and only if w avoids 1324, 24153, 31524, 426153. Examples
321 231 312 213 132 123
w = 213
SLIDE 54 Rook placements on Dw and Bruhat intervals
Theorem (Lewis-M 2013) For all w, #[w, n n 1 . . . 21] p(n, Dw), = if and only if w avoids 1324, 24153, 31524, 426153. Examples
321 231 312 213 132 123
w = 213 w = 1324 p(4, D1324) = 18 #[1324, 4321] = 20
SLIDE 55 q: mq(n, Dw) and #[w, n n 1 · · · 21]
For all w, q(
n 2)inv(w) X
u⌫w
qinv(u) mq(n, Dw)/(q 1)n 2 N[q], = 0 if and only if w avoids 1324, 24153, 31524, 426153. Conjecture (Lewis-Klein-M 2012)
SLIDE 56 q: mq(n, Dw) and #[w, n n 1 · · · 21]
For all w, q(
n 2)inv(w) X
u⌫w
qinv(u) mq(n, Dw)/(q 1)n 2 N[q], = 0 if and only if w avoids 1324, 24153, 31524, 426153. Conjecture (Lewis-Klein-M 2012) Example
321 231 312 213 132 123
w = 213 mq(3, D213) (q 1)3 = q2(q3 + 2q2 + q1)
SLIDE 57 q: mq(n, Dw) and #[w, n n 1 · · · 21]
For all w, q(
n 2)inv(w) X
u⌫w
qinv(u) mq(n, Dw)/(q 1)n 2 N[q], = 0 if and only if w avoids 1324, 24153, 31524, 426153. Conjecture (Lewis-Klein-M 2012) Example
321 231 312 213 132 123
w = 213 mq(3, D213) (q 1)3 = q2(q3 + 2q2 + q1) Remark
- same patterns appear in: Gasharov-Reiner 02,
Postnikov 06, Sj¨
SLIDE 58
Outline of talk
I Is mq(n, S)/(q 1)n a polynomial in q?, Is it in N[q]? For which S is it in N[q]? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw? ⇤ if w avoids 2143 (vexillary) ⇤ if w avoids 1324, 24153, 31524, 426153 II When are rook placements on S related to Bruhat intervals? – If S is a Ferrer’s board: Fλ ? – If S is a diagram of a permutation: Dw? ⇤ if w avoids 1324, 24153, 31524, 426153
⇥ X X
conjecture
X X X X
conjecture
SLIDE 59 Bibliography: https://sites.google.com/site/matrixfinitefields/ Code (sage and maple):
- with A. Klein, J. Lewis, Counting matrices over finite fields with
support on skew Young and Rothe diagrams, arXiv:1203.5804
- with J. Lewis, R. Liu, G. Panova, S. Sam, Y. Zhang, Matrices with
restriced entries and q-analogues of permutations, arXiv:1011.4539
