Rationality, irrationality, and Wilf equivalence in generalized - PowerPoint PPT Presentation

Rationality, irrationality, and Wilf equivalence in generalized factor order Sergey Kitaev Institute of Mathematics, Reykjav´ ık University, IS-103 Reykjav´ ık, Iceland, sergey@ru.is Jeffrey Liese Department of Mathematics, UCSD, La Jolla, CA 92093-0112. USA, jliese@math.ucsd.edu Jeffrey Remmel Department of Mathematics, UCSD, La Jolla, CA 92093-0112. USA, remmel@math.ucsd.edu Bruce E. Sagan Department of Mathematics, Michigan State University East Lansing, MI 48824-1027, sagan@math.msu.edu www.math.msu.edu/ ˜ sagan

Outline

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } .

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w .

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v .

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221.

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221. Factor order is the partial order on P ∗ where w ′ ≤ w iff w ′ is a factor of w .

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221. Factor order is the partial order on P ∗ where w ′ ≤ w iff w ′ is a factor of w . Bj¨ orner found the M¨ obius function of factor order.

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221. Factor order is the partial order on P ∗ where w ′ ≤ w iff w ′ is a factor of w . Bj¨ orner found the M¨ obius function of factor order. Now let ( P , ≤ P ) be a poset.

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221. Factor order is the partial order on P ∗ where w ′ ≤ w iff w ′ is a factor of w . Bj¨ orner found the M¨ obius function of factor order. Now let ( P , ≤ P ) be a poset. Generalized factor order on P ∗ is: u ≤ w if there is a factor w ′ of w with | u | = | w ′ | and u 1 ≤ P w ′ 1 , . . . , u k ≤ P w ′ k where k = | u | .

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221. Factor order is the partial order on P ∗ where w ′ ≤ w iff w ′ is a factor of w . Bj¨ orner found the M¨ obius function of factor order. Now let ( P , ≤ P ) be a poset. Generalized factor order on P ∗ is: u ≤ w if there is a factor w ′ of w with | u | = | w ′ | and u 1 ≤ P w ′ 1 , . . . , u k ≤ P w ′ k where k = | u | . Example. If P is the positive integers then in P ∗ we have 324 ≤ 216541.

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221. Factor order is the partial order on P ∗ where w ′ ≤ w iff w ′ is a factor of w . Bj¨ orner found the M¨ obius function of factor order. Now let ( P , ≤ P ) be a poset. Generalized factor order on P ∗ is: u ≤ w if there is a factor w ′ of w with | u | = | w ′ | and u 1 ≤ P w ′ 1 , . . . , u k ≤ P w ′ k where k = | u | . Example. If P is the positive integers then in P ∗ we have 324 ≤ 216541. Note that generalized factor order becomes factor order if P is an antichain.

Let P be a set and consider the corresponding free monoid P ∗ = { w = w 1 w 2 . . . w k : k ≥ 0 and w i ∈ P for all i } . Let ǫ be the empty word and let | w | denote the length of w . Word w ′ is a factor of w if there are words u , v with w = uw ′ v . Example. w’=322 is a factor of w = 13213221. Factor order is the partial order on P ∗ where w ′ ≤ w iff w ′ is a factor of w . Bj¨ orner found the M¨ obius function of factor order. Now let ( P , ≤ P ) be a poset. Generalized factor order on P ∗ is: u ≤ w if there is a factor w ′ of w with | u | = | w ′ | and u 1 ≤ P w ′ 1 , . . . , u k ≤ P w ′ k where k = | u | . Example. If P is the positive integers then in P ∗ we have 324 ≤ 216541. Note that generalized factor order becomes factor order if P is an antichain. If P = P then factor order is an order on compositions.

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗ If f ∈ Z �� P �� has no constant term, i.e., c ǫ = 0, then define f ∗ = ǫ + f + f 2 + f 3 + · · ·

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗ If f ∈ Z �� P �� has no constant term, i.e., c ǫ = 0, then define f ∗ = ǫ + f + f 2 + f 3 + · · · = ( ǫ − f ) − 1 .

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗ If f ∈ Z �� P �� has no constant term, i.e., c ǫ = 0, then define f ∗ = ǫ + f + f 2 + f 3 + · · · = ( ǫ − f ) − 1 . Then f is rational if it can be constructed from finitely many elements of P using the algebra and star operations.

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗ If f ∈ Z �� P �� has no constant term, i.e., c ǫ = 0, then define f ∗ = ǫ + f + f 2 + f 3 + · · · = ( ǫ − f ) − 1 . Then f is rational if it can be constructed from finitely many elements of P using the algebra and star operations. A language is any L ⊆ P ∗ .

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗ If f ∈ Z �� P �� has no constant term, i.e., c ǫ = 0, then define f ∗ = ǫ + f + f 2 + f 3 + · · · = ( ǫ − f ) − 1 . Then f is rational if it can be constructed from finitely many elements of P using the algebra and star operations. A language is any L ⊆ P ∗ . It has an associated generating function f L = � w ∈L w .

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗ If f ∈ Z �� P �� has no constant term, i.e., c ǫ = 0, then define f ∗ = ǫ + f + f 2 + f 3 + · · · = ( ǫ − f ) − 1 . Then f is rational if it can be constructed from finitely many elements of P using the algebra and star operations. A language is any L ⊆ P ∗ . It has an associated generating function f L = � w ∈L w . The language L is regular if P is finite and f L is rational.

Consider the algebra of formal power series with integer coefficients and the elements of P as noncommuting variables: � � � Z �� P �� = f = c w w : c w ∈ Z for all w . w ∈ P ∗ If f ∈ Z �� P �� has no constant term, i.e., c ǫ = 0, then define f ∗ = ǫ + f + f 2 + f 3 + · · · = ( ǫ − f ) − 1 . Then f is rational if it can be constructed from finitely many elements of P using the algebra and star operations. A language is any L ⊆ P ∗ . It has an associated generating function f L = � w ∈L w . The language L is regular if P is finite and f L is rational. Associated with u ∈ P ∗ is � F ( u ) = { w : w ≥ u } and F ( u ) = w . w ≥ u