Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 1 / 36

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1 Consecutive Patterns: From Permutations to Column-Convex Polyominoes and Back Don Rawlings Mark Tiefenbruck California Polytechnic State University University of California San Luis Obispo, Ca San Diego, Ca Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 1 / 36

2 Problem Sets: Let PS = set of consecutive pattern problems on permutations. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 2 / 36

3 Problem Sets: Let PS = set of consecutive pattern problems on permutations. Let PC = set of consecutive pattern problems on compositions. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 2 / 36

4 Problem Sets: Let PS = set of consecutive pattern problems on permutations. Let PC = set of consecutive pattern problems on compositions. Let PCCP = set of consecutive pattern problems on column-convex polyominoes. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 2 / 36

5 Problem Sets: Let PS = set of consecutive pattern problems on permutations. Let PC = set of consecutive pattern problems on compositions. Let PCCP = set of consecutive pattern problems on column-convex polyominoes. Let PW = set of consecutive pattern problems on words. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 2 / 36

6 Problem Sets: Let PS = set of consecutive pattern problems on permutations. Let PC = set of consecutive pattern problems on compositions. Let PCCP = set of consecutive pattern problems on column-convex polyominoes. Let PW = set of consecutive pattern problems on words. Main Result PS PC PCCP PW. Remark: Our perspective allows powerful methods from the contexts of compositions, column-convex polyominoes, and of words to be applied directly to the enumeration of permutations by consecutive patterns. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 2 / 36

7 Consecutive Patterns in Permutations 6 5 σ = = 2 Examples: σ 1 4 σ 5 = 1 4 is a 12-pattern (ascent) 4 3 Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 3 / 36

8 Consecutive Patterns in Permutations σ = = Examples: σ 4 σ 5 = 1 4 is a 12-pattern (ascent) σ 2 σ 3 σ 4 = is a 231-pattern Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 3 / 36

9 Consecutive Patterns in Permutations 5 6 σ = = Examples: σ 1 4 σ 5 = 1 4 is a 12-pattern (ascent) σ 2 σ 3 σ 4 = is a 231-pattern σ 2 σ 3 σ 4 = and σ 4 σ 5 σ 6 = are peaks (231 or 132-patterns) Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 3 / 36

10 Consecutive Patterns in Permutations 5 6 σ = = Examples: σ 1 4 σ 5 = 1 4 is a 12-pattern (ascent) σ 2 σ 3 σ 4 = is a 231-pattern σ 2 σ 3 σ 4 = and σ 4 σ 5 σ 6 = are peaks (231 or 132-patterns) Definition: For a pattern set P m 1 S m, P(σ) = the total number of times elements of P occur in σ and P no (σ) = the maximum number of non-overlapping times elements of P occur in σ. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 3 / 36

11 Consecutive Patterns in Compositions Notation: K n = {w = w 1 w 2... w n : w 1, w 2,..., w n are positive integers} Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 4 / 36

12 Consecutive Patterns in Compositions Notation: K n = {w = w 1 w 2... w n : w 1, w 2,..., w n are positive integers} Definition: Let sum w = w 1 + w w n. When sum w = m, w is said to be a composition of m into n parts. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 4 / 36

13 Consecutive Patterns in Compositions Notation: K n = {w = w 1 w 2... w n : w 1, w 2,..., w n are positive integers} Definition: Let sum w = w 1 + w w n. When sum w = m, w is said to be a composition of m into n parts. w = = Examples: w 2 w 3 = 7 7 is a level w 2 w 3 w 4 = is a peak (w 2 w 3 > w 4 ) Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 4 / 36

14 Inverse of Fedou s Insertion-Shift Bijection Definitions: For σ S n, set inv i σ = {k : i < k n, σ i > σ k }. Put Λ n = {w K n : w 1 w 2... w n }. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 5 / 36

15 Inverse of Fedou s Insertion-Shift Bijection Definitions: For σ S n, set inv i σ = {k : i < k n, σ i > σ k }. Put Λ n = {w K n : w 1 w 2... w n }. The inverse of Fedou s insertion-shift bijection n : S n Λ n K n is given by the rule n (σ, λ) = w where w i = inv i σ + λ σi. 6 ( , ) = Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 5 / 36

16 σ = = w = = Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 6 /

17 Inverse of Fedou s Insertion-Shift Bijection Definitions: For σ S n, set inv i σ = {k : i < k n, σ i > σ k }. Put Λ n = {w K n : w 1 w 2... w n }. The inverse of Fedou s insertion-shift bijection n : S n Λ n K n is given by the rule n (σ, λ) = w where w i = inv i σ + λ σi. 6 ( , ) = Key Properties: If n (σ, λ) = w, then (1) inv σ + sum λ = sum w and, for i < m, (2) σ i < σ m if and only if w i w m + {j : i < j < m, σ i > σ j }. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 7 / 36

18 Inverse of Fedou s Insertion-Shift Bijection Definitions: For σ S n, set inv i σ = {k : i < k n, σ i > σ k }. Put Λ n = {w K n : w 1 w 2... w n }. The inverse of Fedou s insertion-shift bijection n : S n Λ n K n is given by the rule n (σ, λ) = w where w i = inv i σ + λ σi. 6 ( , ) = Key Properties: If n (σ, λ) = w, then (1) inv σ + sum λ = sum w and, for i < m, (2) σ i < σ m if and only if w i w m + {j : i < j < m, σ i > σ j }. Remark: n preserves general shape. As illustrations, (1) peaks are preserved as σ k < σ k+1 > σ k+2 if and only if w k w k+1 > w k+2 and (2) up-down permutations coincide with up-down compositions. Definition: For P m 1 S m and w K n, set P(w) = P(σ) where σ is the unique permutation satisfying w = n (σ, λ). Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 7 / 36

19 Theorem PS PC If P m 1 S m and if B n S n, then n 0 σ B n q inv σ = n 0 ( p P ) yp p(σ) z n (q; q) n w φ n(b n,λ n) q sum w( p P ) yp p(w) (z/q) n. Moreover, the above equality remains true if y p(σ) p respectively replaced by y pno(σ) p and yp p(w) are and yp pno(w) for some or all p P. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 8 / 36

20 Theorem PS PC If P m 1 S m and if B n S n, then n 0 σ B n q inv σ = n 0 ( p P ) yp p(σ) z n (q; q) n w φ n(b n,λ n) q sum w( p P ) yp p(w) (z/q) n. Moreover, the above equality remains true if y p(σ) p respectively replaced by y pno(σ) p and yp p(w) are and yp pno(w) for some or all p P. Gessel : Carlitz : n 0 n 0 σ UDS n qinv σ zn (q; q) n = sec q z + tan q z w UDK n q sum w (z/q) n = sec q z + tan q z. Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 8 / 36

21 Theorem PS PC If P m 1 S m and if B n S n, then n 0 σ B n q inv σ = n 0 ( p P ) yp p(σ) z n (q; q) n w φ n(b n,λ n) q sum w( p P ) yp p(w) (z/q) n. Moreover, the above equality remains true if y p(σ) p respectively replaced by y pno(σ) p and yp p(w) are and yp pno(w) for some or all p P. Mendes, Remmel : n 0 y pic (σ) qinv σzn σ S n (q; q) n = y 1 y 1 tanq (z y 1) Heubach, Mansour : y 1 y pic (w) q sum w (z/q) n = n 0 y 1 tanq (z y 1) w K n Consecutive Patterns: From Permutations to Column-ConvexAugust Polyominoes 10, 2010 and Back 9 / 36

22 3 rd Ex of Theorem PS PC: For P S m with m > 1, n 0 qinv σ y Pno(σ) zn σ S n (q; q) n = K q (z) 1 y + y (1 z(1 q) 1 ) K q (z) ( σ S n q inv σ 0 P(σ)) z n /(q; q) n is the q-exponential where K q (z) = n 0 generating function for permutations that avoid P. Note: The above is Mendes and Remmel s extension of Kitaev s result. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 10 / 36

23 3 rd Ex of Theorem PS PC: For P S m with m > 1, n 0 qinv σ y Pno(σ) zn σ S n (q; q) n = K q (z) 1 y + y (1 z(1 q) 1 ) K q (z) ( σ S n q inv σ 0 P(σ)) z n /(q; q) n is the q-exponential where K q (z) = n 0 generating function for permutations that avoid P. Note: The above is Mendes and Remmel s extension of Kitaev s result. Application of Theorem PS PC gives y Pno(w) q sum w z n L q (z) = 1 y + y (1 zq(1 q) 1 ) L q (z) n 0 w K n where L q (z) = ( n 0 w K n q sum w 0 P(w)) z n is the generating function for compositions that avoid P. Remark: The latter is more and less general than a result due to Heubach, Kitaev, and Mansour; for a pattern set of cardinality 1, their result holds for an arbitrary alphabet of positive integers. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 10 / 36

24 Consecutive Patterns in Column-Convex Polyominoes Remark 1: The enumeration of CCPs and of subclasses of CCPs by various statistics (area, perimeter, column number,...etc) has been widely studied. However, very little attention has been paid to ridge patterns in CCPs. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 11 / 36

25 Consecutive Patterns in Column-Convex Polyominoes Remark 1: The enumeration of CCPs and of subclasses of CCPs by various statistics (area, perimeter, column number,...etc) has been widely studied. However, very little attention has been paid to ridge patterns in CCPs. Remark 2: The simplest ridge patterns are formed between two adjacent columns. The two-column ridge patterns may be used to characterize many of the common classes of CCPs. For instance, a CCP with no lower descents is known as a directed column-convex polyomino (DCCP). Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 11 / 36

26 Common Classes of CCPs. Directed Column Convex Polyomino (DCCP): No lower descents Parallelogram Polyomino: No lower or upper descents Stack Polyomino: No upper ascents and no lower descents Wall Polyomino: No lower ascents and no lower descents Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 12 / 36

27 Compositions and Wall Polyominoes Notation: Let WP n = the set of wall polyominoes with n columns. Bijection: Let γ n denote the natural bijection from K n to WP n. Example: γ 7 maps the composition w = to Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 13 / 36

28 Compositions and Wall Polyominoes Notation: Let WP n = the set of wall polyominoes with n columns. Bijection: Let γ n denote the natural bijection from K n to WP n. Example: γ 7 maps the composition w = to Properties: If γ n (w) = Q, then area Q = sum w and per Q 2col Q = var w where variation of w is var w = n k=0 w k+1 w k with the convention that w 0 = 0 = w n+1. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 13 / 36

29 Fact PC PCCP If P is a pattern set defined for compositions and if B n K n, then c var w q sum w z n n 0 w B n p P y p(w) p = n 0 c per Q q area Q (z/c 2 ) n p P Q γ n(b n) y p(q) p. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 14 / 36

30 Example of Fact PC PCCP = Raw, Tief : c 2 h n 0 1 a u n 1 au uasc Q Q DCCP a lasc Q l (c 2 qz) n+1 n 1 c 2 hq n+1 k=1 (c 2 qz) n 1 q n n k=1 b ulev Q u b llev Q l c per Q d udes Q h relh Q q area Q z col Q ( )( b l + a l c 2 hq k b 1 c 2 hq k u + c2 dq k ( ) b l + a l c 2 hq k n 1 1 c 2 hq k k=1 ) au 1 c 2 q k 1 q k ( b u + c2 dq k 1 c 2 q k au 1 q k ). Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 15 / 36

31 Example of Fact PC PCCP = Raw, Tief : c 2 h n 0 1 a u n 1 au uasc Q Q DCCP a lasc Q l (c 2 qz) n+1 n 1 c 2 hq n+1 k=1 (c 2 qz) n 1 q n n k=1 b ulev Q u b llev Q l c per Q d udes Q h relh Q q area Q z col Q ( )( b l + a l c 2 hq k b 1 c 2 hq k u + c2 dq k ( ) b l + a l c 2 hq k n 1 1 c 2 hq k k=1 ) au 1 c 2 q k 1 q k ( b u + c2 dq k 1 c 2 q k au 1 q k ). Setting a l = 0, a u = a, b u = b, b l = h = 1, and replacing z by z/c 2 gives the gen func for compositions by ascents, levels, descents, and variation. (c = 1 is a classic result due to Carlitz) a asc w b lev w d des w c var w q sum w z n n 0 w K n c 2 ( ) (qz) n+1 n n 0 1 c 2 q n+1 k=1 b + c2 dq k a 1 c 2 q k 1 q k = a ( ). (qz) n n 1 n 1 1 q n k=1 b + c2 dq k a 1 c 2 q k 1 q k Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 15 / 36

32 The Inclusion PCCP PW { ( Let X = j ) } m : j, m are positive integers and let Y = {( ) } j1 j 2... j n X n : m n = 1 and j k +j k+1 > m k for 1 k < n. m 1 m 2... m n n 0 For a CCP Q with n columns, define δ(q) = ( j 1 j 2... j n ) m 1 m 2... m n where jk is the number cells in Q k, m n = 1, and, for 1 k < n, m k is the change in the y-ordinate from the bottom edge of (k + 1) st column of Q to the top edge of the k th column of Q. δ = ( ) Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 16 / 36

33 Factors and Consecutive Patterns in Words Let X denote the free moniod generated by the alphabet X. Definition: For F X +, a factor f of a word w is a said to be a consecutive F-pattern in w if f F. The number of consecutive F-patterns in w is denoted by F(w). Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 17 / 36

34 Factors and Consecutive Patterns in Words Let X denote the free moniod generated by the alphabet X. Definition: For F X +, a factor f of a word w is a said to be a consecutive F-pattern in w if f F. The number of consecutive F-patterns in w is denoted by F(w). Definition: For F X +, an F-cluster is a triple (w, ν, β) in which w = w 1 w 2... w len w X +, ν = (f (1), f (2),..., f (k) ) for some k 1 with each f (i) F, and β = (b 1, b 2,..., b k ) with each b i being a positive integer where f (i) =w bi w bi w bi +len f (i) 1, each w i w i+1 is a factor of some f (j), b 1 b 2 b k, and if b i = b i+1, then len f (i) <len f (i+1). Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 17 / 36

35 Factors and Consecutive Patterns in Words Let X denote the free moniod generated by the alphabet X. Definition: For F X +, a factor f of a word w is a said to be a consecutive F-pattern in w if f F. The number of consecutive F-patterns in w is denoted by F(w). Definition: For F X +, an F-cluster is a triple (w, ν, β) in which w = w 1 w 2... w len w X +, ν = (f (1), f (2),..., f (k) ) for some k 1 with each f (i) F, and β = (b 1, b 2,..., b k ) with each b i being a positive integer where f (i) =w bi w bi w bi +len f (i) 1, each w i w i+1 is a factor of some f (j), b 1 b 2 b k, and if b i = b i+1, then len f (i) <len f (i+1). Definition: The cluster generating function over a subset W X is the formal series ( ) C F (y, W ) = y f (ν) f w (w, ν, β) C F, w W f F where f (ν) is the number of times f appears as a component in ν. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 17 / 36

36 Words by Factors Theorem (Goulden and Jackson) If, for nonempty L, R X and a nonempty F X +, we define L(y)= l + C F (y, LX ), R(y)= r + C F (y, X R), and l L r R X (y)= x + C F (y, X ) x X and if the result of replacing y f in y by y f 1 is denoted by y 1, then ( ) w = (1 X (y 1)) 1, w X w LX w X R f F ( f F ( f F y f (w) f f Don Rawlings and Mark Tiefenbruck () ) y f (w) f w = L(y 1)(1 X (y 1)) 1, ) y f (w) f w = (1 X (y 1)) 1 R(y 1), and ( ) f (w) y w = C F (y 1, LX R) + L(y 1)(1 X (y 1)) 1 R(y Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 18 / 36

37 Application of Words by Factors Theorem to Permutations Consider the alphabet N = {1, 2, 3,...}, let P m 1 S m, and put D P (y; z) = ( ) yp p(ν) q sum w z len w where p(ν) = f (ν). p P f F p (w,ν,β) C FP Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 19 / 36

38 Application of Words by Factors Theorem to Permutations Consider the alphabet N = {1, 2, 3,...}, let P m 1 S m, and put D P (y; z) = ( ) yp p(ν) q sum w z len w where p(ν) = f (ν). p P f F p (w,ν,β) C FP Replacement of w by q sum w z len w in the first identity of the Words by Factors Theorem and application of Fedou s bijection implies Extension of Rawlings Theorem: If P m 1 S m, then ( n 0 σ S n p P ) yp p(σ) q inv σ z n ( 1 = 1 z(1 q) 1 D P (y 1; z/q)). (q; q) n Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 19 / 36

39 Application of Words by Factors Theorem to Permutations Consider the alphabet N = {1, 2, 3,...}, let P m 1 S m, and put D P (y; z) = ( ) yp p(ν) q sum w z len w where p(ν) = f (ν). p P f F p (w,ν,β) C FP Replacement of w by q sum w z len w in the first identity of the Words by Factors Theorem and application of Fedou s bijection implies Extension of Rawlings Theorem: If P m 1 S m, then ( n 0 σ S n p P ) yp p(σ) q inv σ z n ( 1 = 1 z(1 q) 1 D P (y 1; z/q)). (q; q) n Non-overlapping Version: For P S m with m > 1, then y n 0 σ S n ( Pno(σ) qinv σ 1 zn = 1 z(1 q) 1 (1 y)d P ( 1)). (q; q) n Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 19 / 36

40 Extension of Rawlings Theorem If P m 1 S m, then ( n 0 σ S n p P ) yp p(σ) q inv σ z n ( 1 = 1 z(1 q) 1 D P (y 1; z/q)). (q; q) n Ex 1: Permutations by Peaks Let pic = {132, 231}. As the pic -clusters are in 1-to-1 correspondence with the up-down compositions of odd length > 1, z 1 q + D pic (y; z/q)= 1 y Thus, n 0 ( z y q sum w q w UDK 2n+1 ) 2n+1 = tan q (z y) y. Mendes, Remmel : n 0 y pic (σ) qinv σzn σ S n (q; q) n = y 1 y 1 tanq (z y 1) Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 20 / 36

41 Ex 2: Permutations by Peaks and Twin Peaks Let tpic = {p S 5 : p 1 < p 2 > p 3 < p 4 > p 5 }. Result using extension of Rawlings Theorem: qinv σ x pic (σ) y tpic (σ) zn 1 = (q; q) n 1 z n 0 σ S n 1 q n 1 A n(x 1, y 1)B n (q)(z/q) 2n+1 where A n (x, y)= (xz +yz 2 +xyz 2 )(1 xz xyz xyz 2 yz yz 2 ) 1 z n and B n (q) = tan q z z 2n+1. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 21 / 36

42 Ex 2: Permutations by Peaks and Twin Peaks Let tpic = {p S 5 : p 1 < p 2 > p 3 < p 4 > p 5 }. Result using extension of Rawlings Theorem: qinv σ x pic (σ) y tpic (σ) zn 1 = (q; q) n 1 z n 0 σ S n 1 q n 1 A n(x 1, y 1)B n (q)(z/q) 2n+1 where A n (x, y)= (xz +yz 2 +xyz 2 )(1 xz xyz xyz 2 yz yz 2 ) 1 z n and B n (q) = tan q z z 2n+1. Alternative Result using Pattern Algebra of Goulden and Jackson: qinv σ x pic (σ) y tpic (σ) zn (q; q) n n 0 σ S n ( = 1 s + sin q (z r + ) 2 r + cos q (z r + ) s sin q (z r ) 2 r cos q (z r ) where r ± = (xy 1 ± D)/2, s ± = 1 ± (2x xy 1)/ D, and D = (xy +1) 2 4x. ) 1 Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 21 / 36

43 q-olivier functions z jn+k Φ j,k (z) = (q; q) n 0 jn+k Examples: Φ 1,0 (z) = e q (z), Φ 2,0 (iz) = cos q z, and Φ 2,1 (iz) = i sin q z. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 22 / 36

44 Ex 3: Permutations by (i,d)-peaks and Inversions Let P i,d = {p S i+d 1 : p 1 < p 2 < < p i > p i+1 > > p i+d 1 }. Example of a (3,3)-peak: 5 4 p = = Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 23 / 36

45 Ex 3: Permutations by (i,d)-peaks and Inversions If, for i, j, d 2, we set µ = i + d 2 and ξ m = m 1, then ( (σ) qinv σzn = 1 z(1 q) 1 K i,i,d;1( µ ) y 1 z) 1 (q; q) n µ y 1 y Pi,d n 0 σ S n where, for k 1, K i,j,d;k (z) = m 0 w K i,d;(j,d) m ;k q sum w z len w. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 24 / 36

46 Ex 3: Permutations by (i,d)-peaks and Inversions If, for i, j, d 2, we set µ = i + d 2 and ξ m = m 1, then ( (σ) qinv σzn = 1 z(1 q) 1 K i,i,d;1( µ ) y 1 z) 1 (q; q) n µ y 1 y Pi,d n 0 σ S n where, for k 1, K i,j,d;k (z) = m 0 w K i,d;(j,d) m ;k q sum w z len w. Moreover, K i,j,d;k (z) satisfies, for d 3 and ν = j + d 2, the recurrence K i,j,d;k (z) = ξ µ ν K i,j+1,d 1;1 (ξ ν z) ( z k (q; q) 1 k + ξν k K j,j+1,d 1;k+1 (ξ ν z) ) 1 + K j,j+1,d 1;1 (ξ ν z) with the initial condition K i,j,2;k (z) = ξ i k j ξ µ k ν K i,j+1,d 1;k+1 (ξ ν z) [ Φ j,i+k (ξ j z) + Φ j,i (ξ j z)φ j,k (ξ j z)/φ j,0 (ξ j z)]. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 24 / 36

47 Permutations by (3,3)-peaks and Inversions y n 0 σ S n where ( P3,3(σ) qinv σzn = 1 z(1 q) 1 K 3,3,3;1( 4 ) y 1 z) 1 (q; q) n 4 y 1 K 3,3,3;1 (z) = K 3,4,2;1(ξ 4 z) ( z(1 q) 1 + ξ 1 4 K 3,4,2;2(ξ 4 z) ) 1 + K 3,4,2;1 (ξ 4 z) +ξ 1 4 K 3,4,2;2(ξ 4 z) with K 3,4,2;1 (z) = Φ 4,3(ξ 4 z)φ 4,1 (ξ 4 z) + Φ 4,4 (ξ 4 z) and Φ 4,0 (ξ 4 z) [ K 3,4,2;2 (z) = ξ4 1 Φ ] 4,3(ξ 4 z)φ 4,2 (ξ 4 z) + Φ 4,5 (ξ 4 z). Φ 4,0 (ξ 4 z) Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 25 / 36

48 Ex 4: Permutations by m-peak Ranges of (i, d)-peaks Corollary If i, d 2, m 1, and ν = i + d 2, then the generating function for permutations by uniform m-peak ranges and inversions is y n 0 σ S n P (i,d) m (σ) q inv σ z n ( = 1 z (q; q) n 1 q ) 1 A n,m (y 1)B n (q)z nν+1 n m where A n,m (y) = yz m (1 z) 1 z yz(1 z m ) z n and B n (q) = K i,i,d;1 (z) z nν+1 with K i,i,d;1 (z) as determined earlier. Remark: For i = d = 2 with y = 0, the above provides a solution to a problem posed by Kitaev of counting permutations that avoid (2m + 1)-reverse-alternating patterns. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 26 / 36

49 Ex 5: Permutations by (i,m)-maxima and Inversions Let p (m) S i+1 with p (m)1 < p (m)2 < < p (m)i and p (m)i+1 = i + 1 m. Example: The (3,2)-maxima pattern in S 3+1 : p (2) = = Remark: Carlitz and Scoville referred to (2, 1) and (2, 2)-maxima as rising and falling maxima. 2 Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 27 / 36

50 Ex 5: Permutations by (i,m)-maxima and Inversions Corollary (Words by Factors via extension of Rawlings Theorem): If i 2, 1 m i and ξ i = i 1, then ( i n 0 σ S n m=1 Φ i,k (y 1,..., y i ; z)= (q; q) n 0 in+k ) y p (m)(σ) q inv σ z n ( m = 1 Φ ) i,1(y 1; ξ i z) 1 where (q; q) n ξ i Φ i,0 (y 1; ξ i z) z in+k n 1 ( i 1 [ ]) y i + (y i y m )q m ij +k +m 1. m j=0 m=1 For i = 2, y 1 = y, and y 2 = 1, above gives Mendes and Remmel s q-analog of Elizalde and Noy s result for permutations by p = 132: y n 0 σ S n ( 132(σ) qinv σzn = 1 (q; q) n n 0 (y 1) n q n z 2n+1 ) 1 (q 2 ; q 2 ) n (1 q 2n+1 )(1 q) n. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 28 / 36

51 An Aside Proof of the generating function for permutations by (i, m)-maxima of the previous slide reveals the generating function for up-down permutations of type (i, i, 2; 1) by (i, m)-maxima: z 1 q + σ UDS i,i,2;1 ( i m=1 y p (m) m ) q inv σ z len σ (q; q) inv σ = Φ i,1(y 1,..., y i 1, 1; ξ i z) ξ i Φ i,0 (y 1,..., y i 1, 1; ξ i z). Setting y 1 = y 2 =... = y i = 1, replacing z by (1 q)z, and letting q 1 gives a result of Carlitz s. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 29 / 36

52 Ex 5: Perms by (i,m)-maxima using the Temperley Method If i 2 and 1 m i, then the generating function for permutations by (i, m)-maxima and inversions is also given by where ( i n 0 σ S n T (b) = m=1 = 1 i 1 m=1 ) y p (m)(σ) q inv σ z n m (q; q) n n 0 1 y i 1 (q; q) i 1 z in+1 n 1 1 q in+1 T (q ik ) n 1 k=0 z in n 1 1 q in (y m 1)q m (q; q) m (q m+1 b; q) i m k=1 T (q ik 1 ) 1 y i 1 (q; q) i 1 (1 qb). Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 30 / 36

53 Ex 6: Permutations by maximal number of non-overlapping P = {1243, 1342, 1432, 2341, 2431, 3421} Corollary where K = y n 0 σ S n Pno(σ) qinv σ zn (q; q) n = K 1 y + y(1 z 1 q )K e q (z)e q ( z) + cos 2 q z + sin 2 q z + 2e q ( z) cos q z +(e q (z) + e q ( z)) sin q z 4e q ( z) cos q z. Remark: K is a q-analog of Kitaev s generating function that enumerates permutations that avoid P = {4312, 4213, 4123, 3214, 3124, 2134}. Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 31 / 36

54 Ex 7: DCCPs by All Five Two-Column Statistics Raw, Tief : au uasc Q Q DCCP c 2 h n 0 = 1 a u n 1 a lasc Q l b ulev Q u (c 2 qz) n+1 n 1 c 2 hq n+1 k=1 (c 2 qz) n 1 q n n k=1 b llev Q l c per Q d udes Q h relh Q q area Q z col Q ( )( b l + a l c 2 hq k b 1 c 2 hq k u + c2 dq k ( ) b l + a l c 2 hq k n 1 1 c 2 hq k k=1 ) au 1 c 2 q k 1 q k ( b u + c2 dq k 1 c 2 q k au 1 q k ). Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 32 / 36

55 Ex 8: CCPs by All Six Two-Column Statistics (Temperley) If we set F (x) equal to au uasc Q Q CCP b ulev Q u d udes Q u a lasc Q l b llev Q l d ldes Q l c per Q q area Q h relh Q x α(q) z col Q where α(q) denotes the area of the last column in Q, then where F (x) = R(x) S(x) T (x) R(1) S(1) 1 T (1) R( 1 h ) S( 1 h ) T ( 1 h ) 1 S(1) 1 T (1) S( 1 h ) T ( 1 h ) 1 R(x) = n 0 z n+1 y(x)y(qx)... y(q n 1 x)r(q n x),, S(x) = n 0 z n+1 y(x)y(qx)... y(q n 1 x)s(q n x), T (x) = z n+1 y(x)y(qx)... y(q n 1 x)t(q n x), and... Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 33 / 36

56 Ex 8 Continued : The Rest of the Formula r(x) = s(x) = qxc 4 h 1 qxc 2 h, q 2 x 2 c 4 ha u a l (1 qx)(1 qxc 2 h) + qxc2 a l b u 1 qx qxc 2 d u a l (1 h)(1 qx), t(x) = qxc2 hd u b l 1 qxh + q 2 x 2 c 4 hd u d l (1 qxh)(1 qxc 2 ) + qxc2 h 2 d u a l (1 h)(1 qxh), y(x) = qxc4 ha u b l 1 qxc 2 h + q 2 x 2 c 6 ha u d l (1 qxc 2 )(1 qxc 2 h) qxc 4 ha u a l (1 qx)(1 qxc 2 h) +c 2 b u b l + qxc4 b u d l 1 qxc 2 c2 b u a l 1 qx c2 d u b l 1 qxh qxc 4 d u d l (1 qxh)(1 qxc 2 ) + c 2 d u a l (1 qx)(1 qxh). Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 34 / 36

57 Ex 9: DCCPs by Upper Valleys A column-segment Q k Q k+1 Q k+2 in a column-convex polyomino Q is said to be a valley provided that Q k Q k+1 is an upper descent and Q k+1 Q k+2 is an upper ascent or an upper level. Corollary of Words by Factors y val(q) q area Q z col Q Q DCCP = n 0 (1 y) n q (n+1)(2n+1) z 2n+1 (q; q) 2n+1 (q; q) 2n n 0 (1 y) n q n(2n+1) z 2n (q; q) 2 2n n 0 (1 y) n q (n+1)(2n+1) z 2n+1. (q; q) 2 2n+1 Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 35 / 36

58 Ex 10: CCPs by Peaks, Area, and Column Number (Temperley) Q CCP y pic (Q) q area Q z col Q = ( zq 1 q + 2z2 q 3 (1 q) 3 )(1 + (1 zq2 (1 q) 2 )(1 + 2zq (1 q) 2 ) zq. ) 2yz2 q 3 (1 q) 2 (1 q) 4 Consecutive Patterns: From Permutations to Column-Convex August Polyominoes 10, 2010and Back 36 / 36