Locally Restricted Compositions Past, Present, Future? Ed Bender with Rod Canfield and Jason Gao Papers in the Electronic Journal of Combinatorics 2005, 2009, 2010 Or on my web page: http://www.math.ucsd.edu/ ∼ ebender/ Click on “Bibliography”.
Examples of Compositions A composition of n is a finite list of positive integers c 1 , . . . , c k (called parts) that sum to n . ◮ Unrestricted compositions. ◮ Carlitz compositions: c i � = c i +1 . ◮ Alternating compositions: c 1 < c 2 > c 3 < c 4 · · · . ◮ Distinct parts: c i � = c j when i � = j . ◮ Partitions: c i ≥ c i +1 . ◮ Pattern avoiding: Carlitz 11; partitions 12; distinct 1 · · · 1. An r -rowed composition can be thought of as an ordinary one: row k is c k , c k + r , c k +2 r , . . . ◮ r -rowed Carlitz: adjacent parts differ. ◮ Sum of each row within 1 of n / r .
Allowed Compositions ◮ Local restrictions : For fixed k ◮ c i , . . . , c i + k must satisfy a condition; ◮ condition may be periodic in i ; ◮ may have different conditions near ends. ◮ Recurrence : Can get from any “internal” string c i , . . . , c i + ℓ to any other. ◮ A gcd=1 condition .
Examples of Compositions A composition of n is a finite list of positive integers c 1 , . . . , c k called parts that sum to n . ◮ Unrestricted compositions. ◮ Carlitz compositions: c i � = c i +1 . ◮ Alternating compositions: c 1 < c 2 > c 3 < c 4 · · · . ◮ distinct parts: c i � = c j when i � = j . ◮ Partitions: c i ≥ c i +1 . ◮ Pattern avoiding : Carlitz 11; partitions 12; distinct 1 · · · 1 . An r -rowed composition can be thought of as an ordinary one: row k is c k , c k + r , c k +2 r , . . . ◮ r -rowed Carlitz: adjacent parts differ. ◮ Sum of each row within 1 of n / r .
Some Things to Study Asymptotically The number of compositions of n . Properties of random compositions of n such as: ◮ number of parts ◮ number of rises ( c i < c i +1 ) ◮ part size distribution ◮ longest run ◮ largest part ◮ number of distinct parts ◮ gap-free probability ◮ number of parts of multiplicity k
Our Research 1. Rod: Can we generalize Carlitz compositions? ◮ Bad streetlight: the standard proof ◮ Good streetlight: an alternate proof 2. Transition matrices and the general case ◮ The simple pole and consequences ◮ Normal distributions 3. An aside: Inequality constraints 4. Current work: Part size distribution and being gap-free 5. Current work: Two-dimensional compositions
Carlitz Streetlights I Consider the Carlitz composition 3 + 1 + 2 + 3 + 1 + 2 = 10. ◮ Get a recurrence by adding parts one at a time: ◮ Variables x and t to keep track of sum of parts and last part. xt ◮ f ( x , t ) = start + f ( x , 1) 1 − xt − f ( x , xt ) A ( x ) ◮ Rearrange and iterate to get f ( x , 1) = 1 − B ( x ). Natural(?) idea: c k − 1 , c k , c k +1 all different. No luck. Can do periodic inequality constraints.
Carlitz Streetlights II ANOTHER APPROACH: Add one to each part: Consecutive differences can be restricted. f ( x , t , L , R ) = f ( x , xt , L ′ , R ′ ) � �� � f ( x , xt , D + , R ′ ) + χ (1 ∈R ) f ( x , xt , L ′ , D − ) + χ (1 ∈L ) t + 1 − f ( x , xt , D + , D − ) t
Transition Matrices—General ◮ Construct : Transition matrix T ( x ), x for sum of parts s ( x ) t T ( x ) k � ◮ GF = � � f ( x ) ◮ Can include variables � y to keep track of other things. ◮ Problems: ◮ Need enough “look back” for local restrictions. ◮ What about periodic situations (e.g., up-down compositions)? ◮ Solution: Transition matrix steps k parts at a time where ◮ k is enough for “look back”. ◮ k is a multiple of the period.
Transition Matrices—Finite Pretend T ( x ) is finite. Assume T ( x ) k strictly positive for all k > K . Then for x near the positive real axis ◮ unique, simple largest magnitude eigenvalue λ ( x ). ◮ λ ( x ) analytic, strictly increasing and unbounded. ◮ T = λ E + B , E is a projection, BE = EB = 0. ⇒ T k = λ k E + B k so g ( x ) GF = 1 − λ ( x ) + h ( x ) Consequently ◮ c n = Ar − n (1 + o ( s n )) where λ ( r ) = 1 and 0 < s < 1. ◮ Other variables: asymptotically jointly normal with mean and covariance proportional to n .
Transition Matrices—Infinite Bill Helton told us infinite is easy—just like finite. Problems T ( x ) k strictly positive for k > K becomes ◮ ( T ( x ) k ) i , j > 0 for all k > K ( i , j ). ◮ Convergence issues with T ( x ) solved by ◮ double counting (simple technical problem) ◮ bounding x from above ( what about λ ( r ) = 1 ? ) Then needed to do some functional analysis. If number of entries in T ( x ) with power of x less than k has a polynomial bound, we have convergence for | x | < 1.
Applications We have c n ∼ Ar − n (1 + O ( s n )) for some 0 < s < 1. ◮ Counting compositions gives r and A accurately. ◮ A part of size k at some “location” has exponential “cost”: ≤ Br − i Br − ( n − i − k ) number = Cr k . c n Ar − n ◮ Implies largest part in almost all compositions is at most about log 1 / r ( n ). ◮ Provides estimate for length of longest run of fixed � a as n → ∞ . Can we do more?
An Additional Restriction I Assume that if a part is large compared to neighbors, then any larger part can be used. It follows that ◮ The largest part is almost surely close to log 1 / r ( n ). ◮ Ditto for number of distinct parts. ◮ The probability of being gap free approaches a periodic function of log 1 / r ( n ). Where does this come from? ◮ Large parts behave asymptotically like independent geometric random variables. ◮ Hitczenko and Knopfmacher studied these in 2005. We can do more.
An Additional Restriction II I said Assume that if a part is large compared to neighbors, then any larger part can be used. This can be weakened. For all partial compositions � a and � c that are “long enough”: S ( � a ,� c ) is the set of integers k such that � a , k ,� c is recurrent. Assume that if S ( � a ,� c ) is infinite, it contains all sufficiently large integers. This is currently being finished and written up. Can we do more? Number of parts of multiplicity k . Longest strictly increasing sequence.
Two-Dimensional Compositions What is a two-dimensional composition? One possibility: The support is a convex polyomino . ◮ Klarner and Rivest studied them in 1974. I did later. ◮ In unrestricted case, most look like a tilted, thin rod. ◮ Transition matrix adds a column at a time. ◮ Transition matrix different at the ends. ◮ The λ ( r ) = 1 problem must be dealt with. We are currently working on this, but it is not clear if it will work out. There are other possibilities. Are any interesting?
Some Problems ◮ What more can be said about locally restricted compositions? ◮ Added restrictions with interesting consequences? (Remember the assumption about large parts with small neighbors.) ◮ Higher dimensional compositions? ◮ Other uses of infinite matrix results?
Locally Restricted Compositions Past, Present, Future? Ed Bender with Rod Canfield and Jason Gao Papers in the Electronic Journal of Combinatorics 2005, 2009, 2010 Or on my web page: http://www.math.ucsd.edu/ ∼ ebender/ Click on “Bibliography”. THANK YOU
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