Asymptotic Normality in Combinatorics

Asymptotic normality in combinatorics

Xi Chen

School of Mathematical Sciences Dalian University of Technology [email protected]

2017 International Conference on Combinatorics Institute of Mathematics, Academia Sinica, Taipei, Taiwan May 19, 2017

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 1 / 35 Contents

1 Asymptotic normality

2 Methods and techniques

3 Narayana numbers

4 Problems and conjectures

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 2 / 35 Outline

1 Asymptotic normality

2 Methods and techniques

3 Narayana numbers

4 Problems and conjectures

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 3 / 35 Normal distribution (Gaussian distribution) A random variable X is normally distributed, write X ∼ N (µ, σ2), if

Z x 2 1 − (t−µ) Prob(X ≤ x) = √ e 2σ2 dt. 2πσ2 −∞ Having achieved fame with his calculation on the orbit of asteroid Ceres (discovered in 1801), Gauss later published a treatise on the subject of celestial orbits. In this work he propounded the method of least squares, and provided theoretical justiﬁcation by assuming that observational measurements are normally distributed about their mean.

probability density function of N (µ, σ2) Carl F. Gauss (1777–1855)

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 4 / 35 De Moivre-Laplace theorem

Let Xi , 1 ≤ i ≤ n, be a sequence of independent 0, 1 random variables Pn with Prob(Xi = 1) = p and Prob(Xi = 0) = q = 1 − p. Let Sn = i=1 Xi . De Moivre-Laplace theorem For any ﬁxed x,

Z x 2 √ 1 − t Prob(Sn < np + x npq) → √ e 2 dt, 2π −∞ as n → ∞.

The mean and variance of Sn are np and npq, respectively. √ X n k n−k Prob(Sn < np + x npq) = p q . √ k k

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 5 / 35 Generalization of de Moivre-Laplace theorem

Let Xi , 1 ≤ i ≤ n, be a sequence of independent 0, 1 random variables Pn with Prob(Xi = 1) = pi . Let Sn = i=1 Xi with mean and variance

n n X 2 X µn = pi , σn = pi (1 − pi ). i=1 i=1

De Moivre-Laplace theorem: generalization

Provided σn → ∞, then for any ﬁxed x,

Z x 2 1 − t Prob(Sn < µn + xσn) → √ e 2 dt, 2π −∞ as n → ∞.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 6 / 35 Berry-Esseen theorem

Let Xi , 1 ≤ i ≤ n, be independent random variables with means µi , 2 3 variances σi , and absolute third central moments ρi = E|Xi − µi | . Let Pn P 2 P 2 Sn = i=1 Xi with mean µ = i µi and variance σ = i σi . Berry-Esseen theorem There exist a constant C such that

Z x 2 Pn 1 − t C i=1 ρi √ 2 Prob(Sn < µ + xσ) − e dt ≤ 3 . 2π −∞ σ

A.C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. AMS, 1941.

C.-G. Esseen, On the liapunoﬀ limit of error in the theory of probability, Arkiv for Matematik, Astronomi och Fysik, 1942.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 7 / 35 Asymptotic normality Let a(n, k) be a double-indexed sequence of nonnegative numbers and Pn p(n, k) = a(n, k)/ j=0 a(n, j) the normalized probabilities. We say that a(n, k) is asymptotically normal by a central limit theorem with mean µn 2 and variance σn if

Z x X 1 −t2/2 lim sup p(n, k) − √ e dt = 0. (1) n→∞ x∈R 2π −∞ k≤µn+xσn Say that a(n, k) is asymptotically normal by a local limit theorem on S if

1 −x2/2 lim sup σnp(n, bµn + xσnc) − √ e = 0. (2) n→∞ x∈S 2π When S = R, the validity of (2) implies that of (1) and e−x2/2 Pn a(n, j) a(n, k) ∼ j=0 , n → ∞, p 2 2πσn

where k = µn + xσn for some ﬁxed x. Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 8 / 35 Outline

1 Asymptotic normality

2 Methods and techniques

3 Narayana numbers

4 Problems and conjectures

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 9 / 35 Method 1: direct approach

Sometimes the enumeration problem, suitably normalized, can be viewed as the distribution of a sum of independent 0, 1 variables. Then the generalization of de Moivre-Laplace theorem is applicable. Example number of cycles in a permutation; number of left-to-right minima in a permutation; number of inversions in a permutation; number of distinct prime divisors (Erd˝os-Kactheorem).

W. Feller, The fundamental limit theorems in probability, Bull. AMS, 1945.

W. Feller, An Introduction to Probability Theory and Its Applications, volume I, 1957.

P. Erd˝os,M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, American Journal of Mathematics, 1940. M. Kac, Enigmas of Chance: An Autobiography, 1985.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 10 / 35 Method 2: RZness

Let [a(n, k)]n,k≥1 be an inﬁnite lower triangular matrix with nonnegative Pn k a(n,k) entries. Let An(x) = a(n, k)x and p(n, k) = P . Then k=0 j a(n,j)

0 X An(1) µn = kp(n, k) = An(1) k and 00 2 X 2 2 An(1) 2 σn = k p(n, k) − µn = + µn − µn. An(1) k Qn In particular, if An(x) = i=1(x + rn,i ) have only negative zeros, then

n n X 1 2 X rn,i µn = , σn = 2 . 1 + rn,i (1 + rn,i ) i=1 i=1

2 It follows that σn ≤ µn, ∀n.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 11 / 35 Central limit theorem

Theorem P k 2 Let An(x) = k a(n, k)x have only real zeros, and let µn, σn be the asso- 2 ciated means and variances. If σn → +∞, then a(n, k) are asymptotically normal by a central limit theorem.

P. Lévy, Sur une propriétéde la loi de poisson relative aux petites probabilités, Soc. Math. de France, Comptes rendus des scéances de l’année,1936.

L.H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Statist., 1967.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 12 / 35 Local limit theorem

Theorem (Bender, 1973) 2 Suppose that a(n, k) satisfy a central limit theorem, and σn → +∞. 1 If a(n, k) is unimodal in k for each n, then a(n, k) satisfy a local limit theorem on S = {x : |x| ≥ ε}, ∀ε > 0; 2 if a(n, k) is log-concave in k for each n, then a(n, k) satisfy a local limit theorem on R.

E.A. Bender, Central and local limit theorems applied to asymptotic enumeration, JCTA, 1973.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 13 / 35 Strongly asymptotic normality

Newton’s inequality Pn k If k=0 ak x has only real zeros, then 1 1 a2 ≥ a a 1 + 1 + , k k−1 k+1 k n − k

and the sequence a0, a1,..., an is therefore log-concave and unimodal.

Deﬁnition P k 2 If An(x) = k a(n, k)x have only real zeros and σn → +∞, then we say that a(n, k) are strongly asymptotically normal.

In this case, a(n, k) satisfy both central and local limit theorems.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 14 / 35 Polynomials with only real zeros

Theorem (Wang-Yeh, 2005) Pn k Suppose that An(x) = k=0 a(n, k)x and a(n, k) satisfy

a(n, k) = (rn + sk + t)a(n − 1, k − 1) + (an + bk + c)a(n − 1, k).

If rb ≥ as and (r + s + t)b ≥ (a + c)s, then An(x) have only real zeros.

Theorem (Liu-Wang, 2007)

Suppose that An(x) satisfy deg An−1(x) ≤ deg An(x) ≤ deg An−1(x) + 1 and 0 An(x) = an(x)An−1(x) + bn(x)An−1(x) + cn(x)An−2(x).

if bn(x), cn(x) ≤ 0 for x ≤ 0, then An(x) have only real zeros.

Wang, Yeh, Polynomials with real zeros and P´olya frequency sequences, JCTA, 2005.

Liu, Wang, A uniﬁed approach to polynomial sequences with only real zeros, AAM, 2007.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 15 / 35 Examples of strongly asymptotically normal numbers Example n binomial coefficients k signless Stirling numbers of the first kind c(n, k) Stirling numbers of the second kind S(n, k) Eulerian numbers A(n, k) matching numbers of a graph Laplacian coefficients of a graph

W. Feller, The fundamental limit theorems in probability, Bull. AMS, 1945.

L.H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Statist., 1967.

L. Carlitz, D.C. Kurtz, R. Scoville, O.P. Stackelberg, Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1972. C.D. Godsil, Matching behaviour is asymptotically normal, Combinatorica, 1981.

Y. Wang, H.-X. Zhang, B.-X. Zhu, Asymptotic normality of Laplacian coeﬃcients of graphs, J. Math. Anal. Appl., accepted.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 16 / 35 Method 3: moments

For a random variable X , the associated moment generating function is deﬁned by tX MX (t) := Ee . t2/2 If X ∼ N (0, 1), then MX (t) = e . Theorem (Curtiss, 1942)

Suppose Xn is a sequence of random variables with distribution functions 2 t2/2 1 R x − t F (x). If M (t) → e for all t, then F (x) → √ e 2 dt for all x. n Xn n 2π −∞

J.H. Curtiss, A note on the theory of moment generating functions, Ann. Math. Statist., 1942.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 17 / 35 Method 3: moments

Example number of partitions into distinct parts number of distinct parts in a random partition coeﬃcients of polynomials of binomial type q-Catalan numbers q-derangements

P. Erd˝os,J. Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J., 1941. G. Szekeres, Asymptotic distribution of the number and size of parts in unequal partitions, Bull. Austral. Math. Soc., 1987.

W.M.Y. Goh, E. Schmutz, The number of distinct part sizes in a random partition, JCTA, 1995.

E.R. Canﬁeld, Central and local limit theorems for the coeﬃcients of polynomials of binomial type, JCTA, 1977.

W.Y.C. Chen, C.J. Wang, L.X.W. Wang, The limiting distribution of the coeﬃcients of the q-Catalan numbers, Proc. AMS, 2008.

W.Y.C. Chen, D.G.L. Wang, The limiting distribution of the q-derangement numbers, European J. Combin., 2010.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 18 / 35 Method 4: singularity analysis

Theorem (Bender, 1973) P n k Let f (x, y) = n,k a(n, k)x y with a(n, k) ≥ 0. Suppose there exist 1 a function A(s) continuous and nonzero near 0, 2 a function r(s) with bounded third derivative near 0, 3 a nonnegative integer m, and 4 positive numbers ε and δ such that

x m A(s) 1 − f (x, es ) − r(s) 1 − x/r(s)

is analytic and bounded for |s| < , and |x| < r(0) + δ. Put µ = −r 0(0)/r(0) and σ2 = µ2 − r 00(0)/r(0). If σ2 6= 0, then a(n, k) are asymptotically normal with mean nµ and variance nσ2.

Bender, Central and local limit theorems applied to asymptotic enumeration, JCTA, 1973.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 19 / 35 Outline

1 Asymptotic normality

2 Methods and techniques

3 Narayana numbers

4 Problems and conjectures

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 20 / 35 Narayana numbers

1 n n The Narayana numbers N(n, k) = n k−1 k , 1 ≤ k ≤ n. Question (Shapiro, 2001) Do the rows of the Narayana triangle approach a normal distribution?

 1   1 1     1 3 1    [N(n, k)]n,k≥1 =  1 6 6 1     1 10 20 10 1   . .  . ..

L. Shapiro, Some open questions about random walks, involutions, limiting distribu- tions, and generating functions, Adv. in Appl. Math. 27 (2001) 585–596.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 21 / 35 Narayana polynomials

1 n n The Narayana numbers N(n, k) = n k−1 k , 1 ≤ k ≤ n.

Narayana polynomials

n n X 1 X n n N (x) = N(n, k)xk = xk n n k − 1 k k=1 k=1 satisfy the recurrence

2 (n + 1)Nn(x) = (2n − 1)(1 + x)Nn−1(x) − (n − 2)(1 − x) Nn−2(x)

2 with N1(x) = x and N2(x) = x + x .

It is well known that Nn(x) have only real zeros.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 22 / 35 Asymptotic normality of Narayana numbers I

n 1 X n n 1 2n N (x) = xk ⇒ N (1) = n n k − 1 k n n + 1 n k=1 n 1 X n n 2n − 1 N0 (x) = k xk−1 ⇒ N0 (1) = n n k − 1 k n n − 1 k=1 n 1 X n n 2n − 2 N00(x) = k(k − 1) xk−2 ⇒ N00(1) = (n − 1) n n k − 1 k n n − 1 k=1

0 00 Nn(1) n + 1 2 Nn (1) 2 (n − 1)(n + 1) µn = = , σn = + µn − µn = → +∞. Nn(1) 2 Nn(1) 4(2n − 1)

Theorem (C-Mao-Wang, 2016) The Narayana numbers are strongly asymptotically normal.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 23 / 35 Asymptotic normality of Narayana numbers II

1 n n 1 2n Narayana numbers N(n, k) = n k−1 k have row sums Cn = n+1 n . Consider n n N(n, k) (n + 1) = k−1 k . C 2n n n n To show the strong asymptotical normality of N(n, k), it suﬃces to show that 2 (k−µn) N(n, k) 1 − 2 ∼ e 2σn p 2 Cn 2πσn 2 for some µn and σn. Assume that k = µn + xσn for some ﬁxed x, where µn and σn are the mean and standard variance of N(n, k) respectively. Then µn = (n + 1)/2 since the symmetry of each row. Recall that 2 Pn k σn ≤ µn, ∀n, since Nn(x) = k=1 N(n, k)x have only real zeros. Then k → (n + 1)/2 as n → ∞.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 24 / 35 Asymptotic normality of Narayana numbers II n It is well known that the binomial coeﬃcients k is strongly asymptotically normal with mean n/2 and variance n/4. Then n 2 n 2 − (k−n/2) ∼ e 2n/4 , k p2πn/4 n 2 n 2 − (k−1−n/2) ∼ e 2n/4 , k − 1 p2πn/4 and 2n 22n ∼ √ . n πn Hence n n (k−(n+1)/2)2 N(n, k) (n + 1) k−1 k 1 − = ∼ e 2n/8 , C 2n p n n n 2πn/8 which implies that N(n, k) is strong asymptotically normal with mean (n + 1)/2 and variance n/8. Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 25 / 35 Outline

1 Asymptotic normality

2 Methods and techniques

3 Narayana numbers

4 Problems and conjectures

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 26 / 35 Stack sorting problem The stack sorting problem introduced by Knuth in the 1960’s was a founding inspiration in the study of permutation patterns. Simultaneously he introduced the notion of pattern containment, deﬁn- ing a class of permutations by a forbidden set, and the enumeration of permutations in such classes.

Donald Knuth TAOCP, 1968

D.E. Knuth, The art of computer programmin, Vol. 1: Fundamental algorithms, Addison-Wesley, London, 1968. Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 27 / 35 Stack sorting problem

A stack is a last-in, ﬁrst-out linear sequence accessed at one end called the top. Items are added and removed from the top end by push and pop oper- ations.

top top

← ←

push pop

A stack sorting makes the output permutation to be 12 ··· n.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 28 / 35 Stack sorting problem Greedy algorithm: only pops a stack if pushing the next input would have violated the increasing property of the stack (read from the top). 132 32 1 32 1 2 Ex. 1 3 ↓ 123 12 1 2 3 3

Theorem (Knuth, 1968) Let p = LnR with L and R respectively denoting the string on the left and right of n. The stack-sorting operation s can be deﬁned recursively by s(p) = s(L)s(R)n.

In the above example p = 132 and s(p) = 123.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 29 / 35 Stack sorting problem

231 31 2 31 2 1 Ex. 2 3 ↓ 213 21 2 1 3 3

Theorem (Knuth, 1968) A permutation π can be sorted by a stack if and only if π avoids 231. 1 2n The number of sortable n-permutations is Cn = n+1 n .

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 30 / 35 t-stack sortable permutations

Given stacks S1, S2,..., St . At any point in the sorting process we may push the next input onto one of the stacks or pop one of the stacks. West’s RULE: examine stacks from S1 to St , locate the ﬁrst stack on which a push can be applied; if there is no such stack then St is popped.

231 31 31 1 1 2 2 2 3 2 3

S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 ↓ 123 12 12 1 1 3 3 2 3 2 3

p is t-stack sortable if and only if st (p) = 12 ··· n. All n-permutations are (n − 1)-stack sortable in this model.

J. West, Permutations with forbidden subsequences and stack sortable permutations, PhD thesis, MIT, 1990. Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 31 / 35 Special cases of t-stack sortable permutations

Let Wt (n)= # t-stack sortable n-permutations.

1 2n W1(n) = n+1 n ; Wn−1(n) = n!; 2(3n)! W2(n) = (n+1)!(2n+1)! ; Wn−2(n) = # all n-permutations that do not end in the string n1.

D.E. Knuth, The art of computer programmin, Vol. 1: Fundamental algorithms, Addison-Wesley, London, 1968.

D. Zeilberger, A proof of Julian West’s conjecture that the number of two-stack- sortable permutations of length n is 2(3n)!/((n + 1)!(2n + 1)!), Discrete Math. 102 (1992) 85–93.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 32 / 35 Special cases of t-stack sortable permutations Let Wt (n, k)= # t-stack sortable n-permutations with k descents.

W1(n, k) = N(n, k);

Wn−1(n, k) = A(n, k); (n+k−1)!(2n−k)! W2(n, k) = k!(n−k+1)!(2k−1)!(2n−2k+1)! ; Wn−2(n, k) = A(n, k) − A(n − 2, k − 1).

We can show that these numbers are all strongly asymptotically normal. Conjecture (B´ona,2002) Pn k The polynomial Wn,t (x) = k=1 Wt (n, k)x has only real zeros for ﬁxed n ≥ 2 and 1 ≤ t ≤ n.

M. B´ona,Symmetry and unimodality in t-stack sortable permutations, JCTA, 2002. Conjecture The t-stack sortable numbers are strongly asymptotically normal.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 33 / 35 Shapiro’s problems Theorem (de Moivre-Laplace) n n n 0 , 1 ,..., n is asymptotically normal.

Theorem (Godsil, 1981) n n−1 n−2 0 , 1 , 2 ,... is asymptotically normal.

Conjecture (Shapiro, 2001) n n−a n−2a 0 , b , 2b ,... is asymptotically normal if 0 ≤ a ≤ b.

This conjecture has been settled by Q.-H. Hou recently.

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 34 / 35 Thanks for your attention!

Xi Chen (DUT) Asymptotic normality in combinatorics May 19, 2017 35 / 35