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1 History f -Decompositions 3-Bin Decompositions Future Directions Distribution of Summands in Generalized Zeckendorf Decompositions Archit Kulkarni, Carnegie Mellon University Umang Varma, Kalamazoo College [email protected], [email protected] (with Philippe Demontigny, Thao Do, and David Moon) Advisor: Steven J Miller SMALL REU, Williams College Young Mathematicians Conference, 2013 The Ohio State University Refs History f -Decompositions 3-Bin Decompositions Future Directions Summary Zeckendorf’s Theorem Our research is inspired by an elegant theorem of Zeckendorf. Theorem Write the Fibonacci numbers as F1 = 1, F2 = 2, Fn = Fn−1 + Fn−2 for n > 2. All natural numbers can be uniquely written as a sum of non-consecutive Fibonacci numbers. Proof of existence uses the greedy algorithm and uniqueness follows by induction. Example 2013 = 1597 + 377 + 34 + 5 = F16 + F13 + F8 + F4 2 Refs 3 History f -Decompositions 3-Bin Decompositions Future Directions Summary Previous research The number of summands in the Zeckendorf decomposition of integers in [Fn , Fn+1 ) converges to the Gaussian distribution as n → ∞. This result extends to non-negative recurrence relations of the form Hn+1 = c1 Hn + c2 Hn−1 + · · · + cL Hn−L+1 , where c1 , c2 , . . . , cL are nonnegative integers and L, c1 , cL are positive. Refs 4 History f -Decompositions 3-Bin Decompositions Future Directions Summary Going the other way Previous work: linear recurrence sequence → notion of legality. Our work: notion of legality → linear recurrence sequence. Refs 5 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples f -Decompositions We focused on constructing sequences from notions of legal decomposition. Refs 6 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples f -Decompositions We focused on constructing sequences from notions of legal decomposition. Many notions of “legal” decompositions can be encoded as f -decompositions. Refs History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples f -Decompositions We focused on constructing sequences from notions of legal decomposition. Many notions of “legal” decompositions can be encoded as f -decompositions. Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Refs History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Example (Base b representation) Base b representation can be interpreted as f -decompositions. For example, consider base 5: {an } = 1, 2, 3, 4, 5, 10, 15, 20, 25, 50, 75, 100, . . . Here, an = 5an−4 . 8 Refs 9 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Example (Zeckendorf Decompositions) Consecutive terms may not be chosen: this is equivalent to saying f (n) = 1 for all n ∈ N. We construct a sequence by appending the smallest number that cannot be decomposed using previous terms. {Fn } = Refs 10 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Example (Zeckendorf Decompositions) Consecutive terms may not be chosen: this is equivalent to saying f (n) = 1 for all n ∈ N. We construct a sequence by appending the smallest number that cannot be decomposed using previous terms. {Fn } = 1, Refs 11 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Example (Zeckendorf Decompositions) Consecutive terms may not be chosen: this is equivalent to saying f (n) = 1 for all n ∈ N. We construct a sequence by appending the smallest number that cannot be decomposed using previous terms. {Fn } = 1, 2, Refs 12 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Example (Zeckendorf Decompositions) Consecutive terms may not be chosen: this is equivalent to saying f (n) = 1 for all n ∈ N. We construct a sequence by appending the smallest number that cannot be decomposed using previous terms. {Fn } = 1, 2, 3, Refs 13 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Example (Zeckendorf Decompositions) Consecutive terms may not be chosen: this is equivalent to saying f (n) = 1 for all n ∈ N. We construct a sequence by appending the smallest number that cannot be decomposed using previous terms. {Fn } = 1, 2, 3, 5, Refs 14 History f -Decompositions 3-Bin Decompositions Future Directions Definition and Examples Definition Let f : N0 → N0 . A sum x = ki=0 ani of terms of {an } is an f -decomposition of x using {an } if for every ani , the previous f (ni ) terms are not in the sum. P Example (Zeckendorf Decompositions) Consecutive terms may not be chosen: this is equivalent to saying f (n) = 1 for all n ∈ N. We construct a sequence by appending the smallest number that cannot be decomposed using previous terms. {Fn } = 1, 2, 3, 5, 8, 13, 21, 34, . . . Refs 15 History f -Decompositions 3-Bin Decompositions Future Directions Analogue to Zeckendorf’s Theorem Theorem For any f : N0 → N0 , there exists a sequence of natural numbers {an }∞ n=0 such that every positive integer has a unique f -decomposition using {an }. Refs History f -Decompositions 3-Bin Decompositions Future Directions Analogue to Zeckendorf’s Theorem Theorem For any f : N0 → N0 , there exists a sequence of natural numbers {an }∞ n=0 such that every positive integer has a unique f -decomposition using {an }. Construction of {an }. Let a0 = 1 and an = an−1 + an−1−f (n−1) (assume an = 1 when n < 0). The proof of existence is by induction. Refs 17 History f -Decompositions 3-Bin Decompositions Future Directions Analogue to Zeckendorf’s Theorem Theorem For any f : N0 → N0 , there exists a sequence of natural numbers {an }∞ n=0 such that every positive integer has a unique f -decomposition using {an }. Construction of {an }. Let a0 = 1 and an = an−1 + an−1−f (n−1) (assume an = 1 when n < 0). The proof of existence is by induction. Consider any x ∈ [am , am+1 ) = [am , am + am−f (m) ). Clearly, x − am ∈ [0, am−f (m) ). Refs 18 History f -Decompositions 3-Bin Decompositions Future Directions Analogue to Zeckendorf’s Theorem Theorem For any f : N0 → N0 , there exists a sequence of natural numbers {an }∞ n=0 such that every positive integer has a unique f -decomposition using {an }. Construction of {an }. Let a0 = 1 and an = an−1 + an−1−f (n−1) (assume an = 1 when n < 0). The proof of existence is by induction. Consider any x ∈ [am , am+1 ) = [am , am + am−f (m) ). Clearly, x − am ∈ [0, am−f (m) ). By the induction hypothesis, x − am has an f -decomposition m−f (m)−1 in {an }n=0 . Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence Theorem If f : N0 → N0 is periodic, then the corresponding {an } satisfies a linear recurrence relation. Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence Theorem If f : N0 → N0 is periodic, then the corresponding {an } satisfies a linear recurrence relation. Using linear algebra, we can show that the subsequence {ai,n } = ai , ai+b , ai+2b , ai+3b , . . . satisfies some linear recurrence relation for all i ∈ {0, 1, 2, . . . , b − 1}. Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence Theorem If f : N0 → N0 is periodic, then the corresponding {an } satisfies a linear recurrence relation. Using linear algebra, we can show that the subsequence {ai,n } = ai , ai+b , ai+2b , ai+3b , . . . satisfies some linear recurrence relation for all i ∈ {0, 1, 2, . . . , b − 1}. By finding a common recurrence relation we can show that the interleaved sequence satisfies a linearly recurrent sequence. Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence We can combine two linear recurrence relations by finding the recurrence relation corresponding to the product of their respective characteristic polynomials. Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence We can combine two linear recurrence relations by finding the recurrence relation corresponding to the product of their respective characteristic polynomials. Example {Fn } = 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . satisfies Fn = Fn−1 + Fn−2 . Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence We can combine two linear recurrence relations by finding the recurrence relation corresponding to the product of their respective characteristic polynomials. Example {Fn } = 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . satisfies Fn = Fn−1 + Fn−2 . {an } = 1, 4, 15, 56, 209, 780, 2911, 10864, 40545, . . . satisfies an = 4an−1 − an−2 . Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence We can combine two linear recurrence relations by finding the recurrence relation corresponding to the product of their respective characteristic polynomials. Example {Fn } = 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . satisfies Fn = Fn−1 + Fn−2 . {an } = 1, 4, 15, 56, 209, 780, 2911, 10864, 40545, . . . satisfies an = 4an−1 − an−2 . pF (x )pa (x ) = (x 2 −x −1)(x 2 −4x +1) = x 4 −5x 3 +4x 2 +3x −1. Refs History f -Decompositions 3-Bin Decompositions Future Directions Linear Recurrence We can combine two linear recurrence relations by finding the recurrence relation corresponding to the product of their respective characteristic polynomials. Example {Fn } = 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . satisfies Fn = Fn−1 + Fn−2 . {an } = 1, 4, 15, 56, 209, 780, 2911, 10864, 40545, . . . satisfies an = 4an−1 − an−2 . pF (x )pa (x ) = (x 2 −x −1)(x 2 −4x +1) = x 4 −5x 3 +4x 2 +3x −1. Both sequences satisfy sn = 5sn−1 − 4sn−2 − 3sn−3 + sn−4 . (sn − sn−1 − sn−2 ) − 4(sn−1 − sn−2 − sn−3 ) + (sn−2 − sn−3 − sn−4 ) = 0 Refs History f -Decompositions 3-Bin Decompositions Future Directions Bins Motivation Previous research only handles linear recurrence relations with nonnegative coefficients. 27 Refs History f -Decompositions 3-Bin Decompositions Future Directions Bins Motivation Previous research only handles linear recurrence relations with nonnegative coefficients. Question Do notions of legal decomposition exist for other sequences? 28 Refs History f -Decompositions 3-Bin Decompositions Future Directions Bins Motivation Previous research only handles linear recurrence relations with nonnegative coefficients. Question Do notions of legal decomposition exist for other sequences? Answer Yes! Some functions f yield sequences {an } that can only be described by linear recurrence relations with at least one negative coefficient. 29 Refs 30 History f -Decompositions 3-Bin Decompositions Future Directions Bins Definition of 3-Bin Decomposition Consider the periodic function {f (n)} = 1, 1, 2, 1, 1, 2, . . . . Refs 31 History f -Decompositions 3-Bin Decompositions Future Directions Bins Definition of 3-Bin Decomposition Consider the periodic function {f (n)} = 1, 1, 2, 1, 1, 2, . . . . We have “bins” of width 3, and a legal decomposition contains no consecutive terms and at most one term from each bin. {an } = 1, 2, 3, 4, 7, 11, 15, 26, 41, 56, 97, 153, 209, 362, 571, . . . Refs 32 History f -Decompositions 3-Bin Decompositions Future Directions Bins Definition of 3-Bin Decomposition Consider the periodic function {f (n)} = 1, 1, 2, 1, 1, 2, . . . . We have “bins” of width 3, and a legal decomposition contains no consecutive terms and at most one term from each bin. {an } = 1, 2, 3, 4, 7, 11, 15, 26, 41, 56, 97, 153, 209, 362, 571, . . . We can show that no linear recurrence relation describing this sequence has all nonnegative coefficients, and that the shortest such relation is an = 4an−3 − an−6 . Thus, these decompositions are out of the scope of previous work. Refs 33 History f -Decompositions 3-Bin Decompositions Future Directions Bins Gaussian Behavior of the Number of Summands Question Consider the number of summands for an integer randomly chosen from the interval [0, a3n ). As n → ∞, what can we say about the distribution? Refs History f -Decompositions 3-Bin Decompositions Future Directions Bins Gaussian Behavior of the Number of Summands Question Consider the number of summands for an integer randomly chosen from the interval [0, a3n ). As n → ∞, what can we say about the distribution? For positive linear recurrence sequences, previous work has shown that the distribution approaches a Gaussian. Can we extend this to our new sequences? Refs 35 History f -Decompositions 3-Bin Decompositions Future Directions Refs Bins Gaussian Behavior of the Number of Summands pn,k 150 000 100 000 50 000 2 4 6 First 10 bins 8 10 k 36 History f -Decompositions 3-Bin Decompositions Future Directions Refs Bins Gaussian Behavior of the Number of Summands pn,k 6 ´ 1010 5 ´ 1010 4 ´ 1010 3 ´ 1010 2 ´ 1010 1 ´ 1010 5 10 First 20 bins 15 20 k 37 History f -Decompositions 3-Bin Decompositions Future Directions Refs Bins Gaussian Behavior of the Number of Summands pn,k 1 ´ 1022 8 ´ 1021 6 ´ 1021 4 ´ 1021 2 ´ 1021 10 20 First 40 bins 30 40 k 38 History f -Decompositions 3-Bin Decompositions Future Directions Refs Bins Gaussian Behavior of the Number of Summands pn,k 6 ´ 1044 5 ´ 1044 4 ´ 1044 3 ´ 1044 2 ´ 1044 1 ´ 1044 20 40 First 80 bins 60 80 k History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Number of summands Let pn,k be the number of integers whose decomposition uses k summands all from the first n bins. Claim pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 We count the number of ways to choose the summands. 39 Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Number of summands Let pn,k be the number of integers whose decomposition uses k summands all from the first n bins. Claim pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 We count the number of ways to choose the summands. Either no summand is chosen from the first bin, or... 40 Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Number of summands Let pn,k be the number of integers whose decomposition uses k summands all from the first n bins. Claim pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 We count the number of ways to choose the summands. Either no summand is chosen from the first bin, or... One of the three summands is chosen from the first bin. Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Number of summands Let pn,k be the number of integers whose decomposition uses k summands all from the first n bins. Claim pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 We count the number of ways to choose the summands. Either no summand is chosen from the first bin, or... One of the three summands is chosen from the first bin. Can’t choose both the last element of the first bin and the first element of the second bin. Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Generating Function Begin with pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 43 Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Generating Function Begin with pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 Let F (x , y ) = P n,k≥0 pn,k x ny k . We can write F (x , y ) = xF (x , y ) + 3xyF (x , y ) − x 2 y 2 F (x , y ) + 1 44 Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Generating Function Begin with pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 Let F (x , y ) = P n,k≥0 pn,k x ny k . We can write F (x , y ) = xF (x , y ) + 3xyF (x , y ) − x 2 y 2 F (x , y ) + 1 Simplifying gives us F (x , y ) = 1 1 − x − 3xy + x 2 y 2 Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Generating Function Begin with pn,k = pn−1,k + 3pn−1,k−1 − pn−2,k−2 Let F (x , y ) = P n,k≥0 pn,k x ny k . We can write F (x , y ) = xF (x , y ) + 3xyF (x , y ) − x 2 y 2 F (x , y ) + 1 Simplifying gives us F (x , y ) = 1 1 − x − 3xy + x 2 y 2 Define gn (y ) = k≥0 pn,k y k to isolate x n terms. Using partial fractions and Taylor series, P √ gn 46 (3y +1+ (y )= ) 5y 2 +6y +1 n+1 2n+1 √ ( √ − 3y +1− 5y 2 +6y +1 ) 5y 2 +6y +1 n+1 Refs 47 History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Mean and Variance We can use gn (y ) = k≥0 pn,k y k to compute mean and variance of the random variable Xn , the number of summands for an integer randomly chosen from [0, a3n ). P Refs 48 History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Mean and Variance We can use gn (y ) = k≥0 pn,k y k to compute mean and variance of the random variable Xn , the number of summands for an integer randomly chosen from [0, a3n ). P The mean is √ 8 + 3 12 gn0 (1) =√ µn = √ n + O(1) gn (1) 12 4 + 12 Refs 49 History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Mean and Variance We can use gn (y ) = k≥0 pn,k y k to compute mean and variance of the random variable Xn , the number of summands for an integer randomly chosen from [0, a3n ). P The mean is √ 8 + 3 12 gn0 (1) =√ µn = √ n + O(1) gn (1) 12 4 + 12 The variance is σn2 = d dy √ [ygn0 (y )] g(1) y =1 2 −µ = 3 n + O(1) 9 Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Moment Generating Function If we normalize Xn to Yn = (Xn − µn )/σn , the moment generating function of Yn is X pn,k e t tYn P MYn (t) = E(e ) = k≥0 50 (k−µn ) σn k≥0 pn,k = gn (e t/σn )e −tµn /σn gn (1) Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Moment Generating Function If we normalize Xn to Yn = (Xn − µn )/σn , the moment generating function of Yn is X pn,k e t tYn P MYn (t) = E(e ) = k≥0 (k−µn ) σn k≥0 pn,k = gn (e t/σn )e −tµn /σn gn (1) After multiple Taylor Series expansions, we get log(MYn (t)) = as n → ∞. 51 t2 + o(1) 2 Refs History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Moment Generating Function If we normalize Xn to Yn = (Xn − µn )/σn , the moment generating function of Yn is X pn,k e t tYn P MYn (t) = E(e ) = k≥0 (k−µn ) σn k≥0 pn,k = gn (e t/σn )e −tµn /σn gn (1) After multiple Taylor Series expansions, we get log(MYn (t)) = t2 + o(1) 2 as n → ∞. Hence, the distribution of Yn converges to the standard normal distribution. 52 Refs 53 History f -Decompositions 3-Bin Decompositions Future Directions Proof of Gaussianity Generalizations This method can easily be generalized for large classes of f -decompositions. We have proved Gaussian behavior in the following cases: every bin is of width b for any b ≥ 3 bins have alternating widths x and y Refs 54 History f -Decompositions 3-Bin Decompositions Future Directions Future Directions Does Gaussian behavior hold for all f -decompositions if f is periodic? Can we find interesting notions of legal decomposition that are more general than f -decompositions? Refs 55 History f -Decompositions 3-Bin Decompositions Future Directions Thanks We would like to thank Our advisor, Prof. Steven Miller. Our co-authors Philippe Demontigny, Thao Do, and David Moon. Everyone involved with the SMALL REU at Williams College. Williams College and the NSF (Grant DMS0850577) for funding our research. Refs 56 History f -Decompositions 3-Bin Decompositions Future Directions References Kologlu, Kopp, Miller and Wang: Gaussianity for Fibonacci case, Fibonacci Quarterly 49 (2011), no. 2, 116–130. S. J. Miller and Y. Wang, From Fibonacci numbers to Central Limit Type Theorems, Journal of Combinatorial Theory, Series A 119 (2012), no. 7, 1398–1413. E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bulletin de la Société Royale des Sciences de Liège 41 (1972), pages 179–182. Refs