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Enumeration of Colored Dyck Paths Via Partial Bell Polynomials

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Enumeration of Colored Dyck Paths Via Partial Bell Polynomials Enumeration of Colored Dyck Paths Via Partial Bell Polynomials Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara and Michael D. Weiner Abstract Weconsideraclassoflatticepathswithcertainrestrictionsontheirascents and down-steps and use them as building blocks to construct various families of Dyck paths. We let every building block Pj take on c j colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in a unified manner. Keywords Colored Dyck paths Colored Dyck words Colored Motzkin paths Partial Bell polynomials · · · 2010 Mathematics Subject Classification Primary: 05A15 Secondary: 05A19 · 1 Introduction A Dyck path of semilength n is a lattice path in the first quadrant, which begins at the origin (0, 0),endsat(2n, 0),andconsistsofsteps(1, 1) and (1, 1).Itiscustomary to encode an up-step (1, 1) with the letter u and a down-step (1−, 1) with the letter − D. Birmajer Department of Mathematics, Nazareth College, 4245 East Ave., Rochester, NY 14618, USA e-mail: [email protected] J. B. Gil (B) M. D. Weiner Penn State Altoona,· 3000 Ivyside Park, Altoona, PA 16601, USA e-mail: [email protected] M. D. Weiner e-mail: [email protected] P. R. W. McNamara Department of Mathematics, Bucknell University, 1 Dent Drive, Lewisburg, PA 17837, USA e-mail: [email protected] ©SpringerNatureSwitzerlandAG2019 155 G. E. Andrews et al. (eds.), Lattice Path Combinatorics and Applications, Developments in Mathematics 58, https://doi.org/10.1007/978-3-030-11102-1_8 [email protected] 156 D. Birmajer et al. d.Thus,everyDyckpathcanbeencodedbyacorrespondingDyck word of u’s and d’s. We will freely pass from paths to words and vice versa. Much is known about Dyck paths and their connection to other combinato- rial structures like rooted trees, noncrossing partitions, polygon dissections, Young tableaux, and other lattice paths. While there is a vast literature on the enumeration of Dyck paths and related combinatorial objects according to various statistics, for the scope of the present work, we only refer to the closely related papers [1, 7, 10]. For more information, the reader is referred to the general overview on lattice path enumeration written by Krattenthaler in [4,Chap.10]. For a, b N0 N 0 with a b 0andc (c1, c2,...)with c j N0,we define ∈ = ∪{ } + ̸= = ∈ c Dn(a, b) as the set of Dyck words of semilength (a b)n created from strings of the form (a b) j b( j 1) 1 + P0 “d ”andPj “u + d − + ”for j 1,...,n, such that each maximal (a b) j- = =(a b) j = + ascent substring u + may be colored in c j different ways. We use c j 0if(a b) j- c = + ascents are to be avoided. We will refer to the elements of Dn(a, b) as colored Dyck paths or colored Dyck words. Note that if a 1, b 0, and c is the sequence of ones c 1 (1, 1,...), = = j = = then the building blocks take the form P0 “d ”, Pj “u d ”for j 1,...,n,and 1 = = = Dn (1, 0) is just the set of regular Dyck words of semilength n. c In this paper, we are interested in counting the number of elements in Dn(a, b). c For the sequence given by yn Dn(a, b) ,weprovearecurrencerelationofcon- volution type (see Theorem 1)andgivearepresentationof= y in terms of partial Bell ! ! n polynomials in the elements of the! sequence! c (c1, c2,...)(see Theorem 2). We conclude with several examples that illustrate= the use of our formulas for various values of the parameters a and b as well as some interesting coloring choices. 2 Enumeration of Colored Dyck Words c Our technique for enumerating Dn(a, b) will be to show in Theorem 1 and c Proposition 1 that the sequence yn D (a, b) satisfies the same initial condition = n and recurrence relation as a sequence (zn) involving Bell polynomials. As a direct ! ! c consequence, we get the promised enumeration! ! of Dn(a, b) in terms of partial Bell polynomials (Theorem 2). Theorem 1 For a , b N0 with a b 0 and c (c1, c2,...) with c j N0,let ∈ + ̸= = c ∈ (yn) be the sequence defined by y0 1 and yn Dn(a, b) for n 1.Then,yn satisfies the recurrence = = ≥ ! ! ! ! n yn cℓ yi1 yiaℓ b , (8.1) = ··· + ℓ 1 i1 iaℓ b n ℓ "= +···+"+ = − where each i j is a nonnegative integer. [email protected] Enumeration of Colored Dyck Paths Via Partial Bell Polynomials 157 Proof We will prove (8.1)byshowingthatthereisabijectionbetweenthesetsof objects counted by each side of the equation. The left-hand side counts colored Dyck words of semilength (a b)n.Theright-handsidecountstuplesoftheform + (ℓ, C D1, D2,...,Daℓ b), ; + where 1 ℓ n, • ≤ ≤ C is a color from a choice of cℓ colors, • D is a member of Dc (a, b),i.e.,acoloredDyckwordofsemilength(a b)i , j i j j • and + i1 iaℓ b n ℓ. • +···+ + = − From this tuple, we will construct a colored Dyck word w of semilength (a b)n in the following fashion. + Due to the ℓ and C appearing in the tuple, we begin with the string v (a b)ℓ b(ℓ 1) 1 (a b)ℓ = u + d − + with the substring u + colored C.WethenappendD1, D2, …, Daℓ b to v,separatingeachadjacentpair(Di , Di 1) by an additional copy of the letter+ d.Inthisway,astringw is obtained. We need+ to check that this map is well defined, meaning that w is a colored Dyck word of semilength (a b)n. Let us first check that w contains equal numbers of the letters u+and d.Sincethe Di already satisfy this condition, we need (a b)ℓ (b(ℓ 1) 1) (aℓ b 1), + = − + + + − which is true. Similar reasoning shows the “Dyck” property, i.e., that any prefix of w has at least as many appearances of u as of d.Todeterminethesemilengthofw, we count the number of appearances of u as (a b)ℓ (a b)(n ℓ) (a b)n, + + + − = + as desired. By construction, each maximal (a b) j-ascent has an appropriate color and we conclude that w is a colored Dyck word+ of semilength (a b)n. To show that this map f from the tuple to w is a bijection, we+ argue that it has a well-defined inverse g.Thus,letw be a colored Dyck path of semilength (a b)n, and recall that a and b are fixed. The length L and color of the ascent sequence+ at L 1 the beginning of w determine ℓ a b and C at the start of the tuple g(w).Letw = + (a b)ℓ b(ℓ 1) 1 denote the word obtained from w by removing this prefix u + d − + from w. See Fig. 1 for a schematic example. We next wish to determine D1,...,Daℓ b from w1.Letussaythatw1 has starting height aℓ b 1, meaning that it has this+ many more copies of d than of u.Noticethatthisstartingheightisnonnegative.+ − To determine D1,weproceedbyfindingthesmallestr such that the suffix of w1 corresponding to those letters strictly after position r has starting height one less 1 2 than that of w .Then,weletw be that suffix of w and we let D1 be the prefix of w1 corresponding to those letters strictly before position r.Inparticular,ifthefirst [email protected] 158 D. Birmajer et al. D1 D2 r D3 D4 D5 w2 w1 Fig. 1 A schematic example of determining (ℓ, C D1, D2,...,Daℓ b) from w as in the proof of Theorem 1, where the semicircles represent colored; Dyck paths. We+ have L 5, a 5, b 0, i = = = ℓ 1, and D3 is an empty word. For i 1, 2, we see that w is the portion of w to the right of the corresponding= dashed line = 1 2 letter of w is d,thenwegetthatD1 is the empty word, and we let w be the word 1 1 obtained from w by deleting this 1-letter prefix. If the first letter of w is u,thenD1 will be nonempty. If no such position r exists, then it must be the case that w1 has 1 excess 0 and we let D1 w . = By the definition of r, D1 has equal numbers of u’s and d’s, and it satisfies the Dyck property. Moreover, every maximal (a b) j-ascent sequence is immediately followed by a (b( j 1) 1)-descent sequence+ because w has this property, and because these ascent− lengths+ (a b) j are at least as large as their partnering descent + lengths (b( j 1) 1). In other words, D1 is a colored Dyck word. We continue − + 3 s in this exact manner to determine the full sequences w ,...,w and D1,...,Ds i i 1 for some s.Sinceeachw has starting height one less than w − ,wededucethat s aℓ b,asdesired. = + Notice that the resulting tuple g(w) (ℓ, C D1, D2,...,Daℓ b) satisfies the properties in the four bullet points given at= the beginning; of this proof.+ In particular, since w has semilength (a b)n,thetotalnumberofu’s in D1, D2,...,Daℓ b + + equals (a b)n ℓ(a b),andsoi1 iaℓ b n ℓ.Weconcludethatg maps colored+ Dyck− words+ of semilength+···+(a b)+n to= tuples− of the desired type. Finally, one can readily observe that g f and+f g both equal the identity map.
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