Pattern avoidance in rook monoids Lara Pudwell Pattern avoidance Definitions in rook monoids Rook Monoids Avoidance 1d Avoidance All 0/No 0 Dan Daly (Southeast Missouri State University) patterns Other patterns Lara Pudwell (Valparaiso University) 2d Avoidance Connections to other objects Special Session on Patterns in Permutations and Words Conclusion Joint Mathematics Meetings 2013 San Diego, California January 12, 2013
Rook Monoids Pattern avoidance in rook monoids Definition Lara Pudwell Let n ∈ N . The rook monoid R n is the set of all n × n Definitions Rook Monoids { 0 , 1 } -matrices such that each row and each column contains Avoidance at most one 1. 1d Avoidance All 0/No 0 patterns Example members of R 7 : Other patterns 2d Avoidance 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Connections 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 to other objects 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Conclusion 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 Notice: n × n permutation matrices are a submonoid of R n .
Rook Placements Pattern avoidance in rook monoids Lara Pudwell Have an n × n grid. Definitions Place k rooks (0 ≤ k ≤ n ) in non-attacking position. Rook Monoids Avoidance (No more than one rook in each row, no more than one 1d Avoidance rook in each column). All 0/No 0 patterns Other patterns 2d Avoidance Connections to other objects Conclusion
Rook Polynomials R n ( x ) = � n k =0 r n , k x k where r n , k is the number of placements Pattern avoidance in rook monoids of k rooks on an n × n board. Lara Pudwell Definitions Rook Monoids Avoidance 1d Avoidance All 0/No 0 patterns Other patterns 2d Avoidance Connections to other objects Conclusion
Rook Polynomials R n ( x ) = � n k =0 r n , k x k where r n , k is the number of placements Pattern avoidance in rook monoids of k rooks on an n × n board. Lara Pudwell Definitions R 1 ( x ) = x + 1 Rook Monoids Avoidance 1d Avoidance All 0/No 0 patterns Other patterns R 2 ( x ) = 2 x 2 + 4 x + 1 2d Avoidance Connections to other objects Conclusion R 3 ( x ) = 6 x 3 + 18 x 2 + 9 x + 1
Rook Polynomials R n ( x ) = � n k =0 r n , k x k where r n , k is the number of placements Pattern avoidance in rook monoids of k rooks on an n × n board. Lara Pudwell Definitions R 1 ( x ) = x + 1 Rook Monoids Avoidance 1d Avoidance All 0/No 0 patterns Other patterns R 2 ( x ) = 2 x 2 + 4 x + 1 2d Avoidance Connections to other objects Conclusion R 3 ( x ) = 6 x 3 + 18 x 2 + 9 x + 1 � n � 2 k !. In general r n , k = k
A new enumeration problem Pattern avoidance in rook monoids Known: How many ways can we place k rooks on an n × n Lara Pudwell grid? � n � 2 k ! Definitions r n , k = Rook Monoids k Avoidance n ! = e ( 1 − x ) x ∞ R n (1) x n 1d Avoidance � All 0/No 0 1 − x patterns Other patterns n =0 Sequence: 2 , 7 , 34 , 209 , 1546 , 13327 , . . . (OEIS A002720) 2d Avoidance Connections to other objects Conclusion New question: How many ways can we place k rooks on an n × n grid so they avoid a given smaller rook placement pattern?
Rook Strings Pattern avoidance in rook monoids Given an n × n rook placement, associate a string r 1 · · · r n such Lara Pudwell that: Definitions Rook Monoids If there is a rook in column i , row j , then r i = j . Avoidance 1d Avoidance If column i is empty, then r i = 0. All 0/No 0 patterns Other patterns 2d Avoidance Connections to other objects Conclusion 2473156 0000000 3105006
Rook string avoidance Pattern avoidance in Definition rook monoids Given a rook pattern q ∈ R m and any element r ∈ R n , r Lara Pudwell contains q if there exist 1 ≤ i 1 < · · · < i m ≤ n such that: Definitions Rook Monoids Avoidance q j = 0 if any only if r i j = 0 1d Avoidance The nonzero members of r i 1 · · · r i n are order-isomorphic to All 0/No 0 patterns the non-zero enties of q . Other patterns 2d Avoidance Otherwise r avoids q . Connections to other objects Conclusion Example: 3402 ∈ R 4 contains 0, 1, 01, 10, 12, 21, 201. avoids 102.
Notation Pattern R n ( q ) = { r ∈ R n | r avoids q } avoidance in rook monoids R n , k ( q ) = { r ∈ R n | r avoids q , r has k nonzero entries } Lara Pudwell r n ( q ) = |R n ( q ) | Definitions Rook Monoids r n , k ( q ) = |R n , k ( q ) | Avoidance 1d Avoidance All 0/No 0 patterns For example: Other patterns 2d Avoidance R 2 (01) = { 00 , 10 , 20 , 12 , 21 } Connections to other R 2 , 0 (01) = { 00 } objects Conclusion R 2 , 1 (01) = { 10 , 20 } R 2 , 2 (01) = { 12 , 21 } r 2 (01) = 5, r 2 , 0 (01) = 1, r 2 , 1 (01) = 2, r 2 , 2 (01) = 2
The pattern 0 · · · 0 Pattern avoidance in rook monoids Lara Pudwell Definitions r avoids 0 · · · 0 Rook Monoids Avoidance � �� � j 1d Avoidance ⇐ ⇒ r has at most j − 1 0s. All 0/No 0 patterns Other patterns ⇐ ⇒ r has at least n − j + 1 nonzero entries. � � n � 2 k ! 2d Avoidance r n , k = k ≥ n − j + 1 k r n , k (0 · · · 0 ) = Connections to other � �� � 0 k < n − j + 1 objects j Conclusion
The pattern 0 · · · 0 Pattern avoidance in n � n � 2 rook monoids � r n (0 · · · 0 ) = k ! Lara Pudwell k � �� � k = n − j +1 j Definitions Rook Monoids Avoidance In particular: 1d Avoidance � n � 2 k ! = n ! r n (0) = � n All 0/No 0 k = n k patterns r n (00) = � n � n � 2 k ! = ( n + 1)! Other patterns k = n − 1 k 2d Avoidance Connections to other objects Conclusion
The pattern 0 · · · 0 Pattern avoidance in n � n � 2 rook monoids � r n (0 · · · 0 ) = k ! Lara Pudwell k � �� � k = n − j +1 j Definitions Rook Monoids Avoidance In particular: 1d Avoidance � n � 2 k ! = n ! r n (0) = � n All 0/No 0 k = n k patterns r n (00) = � n � n � 2 k ! = ( n + 1)! Other patterns k = n − 1 k 2d Avoidance Connections to other In general for fixed j objects Conclusion j ∞ ) x n x i − 1 � � r n (0 · · · 0 n ! = ( i − 1)!(1 − x ) i � �� � n =0 i =1 j
Permutation patterns Pattern avoidance in rook monoids Consider ρ ∈ S j . Lara Pudwell n � n � 2 s k ( ρ ) and r n ( ρ ) = � � n � 2 s k ( ρ ) Then r n , k ( ρ ) = Definitions k k k =0 Rook Monoids Avoidance 1d Avoidance All 0/No 0 patterns Other patterns 2d Avoidance Connections to other objects Conclusion
Permutation patterns Pattern avoidance in rook monoids Consider ρ ∈ S j . Lara Pudwell n � n � 2 s k ( ρ ) and r n ( ρ ) = � � n � 2 s k ( ρ ) Then r n , k ( ρ ) = Definitions k k k =0 Rook Monoids Avoidance 1d Avoidance All 0/No 0 We have: patterns n Other patterns � n � 2 s k (1) = � n � � r n (1) = s 0 (1) = 1 2d Avoidance 0 k k =0 Connections n � 2 = � n � 2 n � � to other r n (12) = r n (21) = (OEIS A000984) objects k n k =0 Conclusion For ρ ∈ S 3 , � 2 k � n � � n � 2 C k where C k = k r n ( ρ ) = ( k + 1) (OEIS A086618) k k =0
Small patterns Pattern Rook patterns of length 3 or less include: avoidance in rook monoids 0,1 Lara Pudwell 00, 01, 10, 12, 21 Definitions Rook Monoids 000, 001, 010, 100, 012, 102, 120, 021, 201, 210, 123, Avoidance 132, 213, 231, 312, 321 1d Avoidance All 0/No 0 patterns Other patterns 2d Avoidance Connections to other objects Conclusion
Small patterns Pattern Rook patterns of length 3 or less include: avoidance in rook monoids 0,1 Lara Pudwell 00, 01, 10, 12, 21 Definitions Rook Monoids 000, 001, 010, 100, 012, 102, 120, 021, 201, 210, 123, Avoidance 132, 213, 231, 312, 321 1d Avoidance All 0/No 0 patterns Other patterns We have seen how to enumerate patterns with all 0s and 2d Avoidance patterns with no zeros. Connections to other objects Conclusion
Small patterns Pattern Rook patterns of length 3 or less include: avoidance in rook monoids 0,1 Lara Pudwell 00, 01, 10, 12, 21 Definitions Rook Monoids 000, 001, 010, 100, 012, 102, 120, 021, 201, 210, 123, Avoidance 132, 213, 231, 312, 321 1d Avoidance All 0/No 0 patterns Other patterns We have seen how to enumerate patterns with all 0s and 2d Avoidance patterns with no zeros. Connections to other objects Conclusion r n ( p ) = r n ( q ) if rook placement p can be obtained from q by the action of the dihedral group on the n × n square (then reducing non-zero entries).
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