Between Broadway and the Hudson: A Bijection of Corridor Paths
aa r X i v : . [ m a t h . C O ] J u l BETWEEN BROADWAY AND THE HUDSON:A BIJECTION OF CORRIDOR PATHS
NACHUM DERSHOWITZSCHOOL OF COMPUTER SCIENCETEL AVIV UNIVERSITYRAMAT AVIV, ISRAEL
Canal street, running across Broadway to the Hudson, near the centre of the city,is a spacious street, principally occupied by retail stores. . . .The streets are generally well paved, with good side walks,lighted at night with lamps, and some of them supplied with gas lights.—
The Treasury of Knowledge, and Library of Reference (1834)
Abstract.
We present a substantial generalization of the equinumeracy of grand Dyckpaths and Dyck-path prefixes, constrained within a band. The number of constrained pathsstarting at level i and ending in a window of size 2 j + j and ending in a window of size 2 i + Introduction
We are interested in enumerating lattice paths that remain within a band of height h ,sometimes called corridor paths [1]. Sort of like walking in Manhattan, sticking west ofBroadway (Figure 1).Let i n ⟿ h ℓ , or just i n ⟿ ℓ (fixing h ), denote the number of monotonic lattice paths from ⟨ , i ⟩ to ⟨ n, ℓ ⟩ with n steps that stay within (but may touch) the boundaries y = y = h ,for some given (maximum) height h . Let H = [ ∶ h ] be the ordinate bounds within whichsteps are permissible. Steps are diagonal, NE (northeast, ↗ ), taking ⟨ x, y ⟩ ↦ ⟨ x + , y + ⟩ ,and SE (southeast, ↘ ), taking ⟨ x, y ⟩ ↦ ⟨ x + , y − ⟩ , both with the proviso that the newordinate position y ± ∈ H , as the case may be. It is easy to see that one always has n + i ≡ ℓ ( mod 2 ) , or else there are zero n -step paths starting at level i and ending at ℓ . SeeFigure 2 for a sample path in 1 ⟿ Date : July 17, 2020. Height here is the maximum length of a unidirectional path (just NE or just SE). Some might prefer tosay that the width of the corridor is h +
1, since h + igure 1. Manhattan neighborhoods, with East-West streets andNorth-South avenues, bounded by Broadway on the East and theHudson River on the West, with Union Square serving as ori-gin. (Image © Hagstrom Map Company, Inc., in the public domainat .)The basic recurrence is i n ⟿ ℓ = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ i ∉ H or ℓ ∉ H [ i = ℓ ] if n = ( i n − ⟿ ℓ − ) + ( i n − ⟿ ℓ + ) otherwisewhere the bracketed condition [ i = ℓ ] is Iverson’s notation for a characteristic function (1when true; 0 when false), and the conditions are taken in order. h = k = i = ℓ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = J Figure 2.
A diagonal path (counted by) 1 ⟿
3, which goes from i = ℓ = h =
4. The target region J is J ± K = [ ∶ ] . Its center is k = j = k and 1 above). Figure 3.
A right-left version of the constrained path 1 ⟿
3, consisting of7 right ( + ) steps (colored blue) and 5 left ( − ) steps (red) along a 5-vertexpoint graph P (labeled 0,1,2,3,4), starting at vertex i =
1. The path in thisrepresentation is ++−++−−−−+++ , based at 1. It is an accordion fold of theblue path in Figure 2. The green vertical line serves as a “center of attraction”in Section 5 and Figure 5.The ends of the paths we are interested in fall within a range, J , not just a single point ℓ .For example, the window J = [ ∶ ] has 6 possible landing spots, but only half of them arefeasible, depending on whether n + i is odd or even. Only those ℓ ∈ J with the same parityas n + i are relevant. Our goal is to count i n ⟿ J = ∑ ℓ ∈ J i n ⟿ ℓ = ∑ ℓ ∈ Jℓ ≡ n + i ( mod 2 ) i n ⟿ ℓ the number of paths constrained to any corridor H = [ ∶ h ] and ending at any (feasible)ordinate in the window J .These constrained lattice paths are equivalent to walks along a path graph, forward andbackward. See Figure 3. When i = J = H (anywhere), walks for h = , , , , , , , , ⌊ n / ⌋ ), A000045 (Fibonacci), A038754 ( { , } n ), A028495, A030436, A061551, s = t = ⟨ a, b ⟩ Figure 4.
An orthogonal path, starting at the origin and ending at ⟨ a, b ⟩ = ⟨ , ⟩ , consisting of 7 N-steps and 5 E-steps, staying strictly within bounds s = y = x +
4) and t = y = x − in Sloane’s Encyclopedia of Integer Sequences (OEIS) [16]. Thefollowing sequence for h = n = , , , . . . , does not appear (yet!): 1, 2, 3, 6, 10, 20, 35, 70,126, 251, 460, 911, 1690, 3327, 6225, 12190, 22950, 44744, 84626, 164407, . . . . But its oddelements are enumerated at A216710, while its even ones are A224514.Such paths in a path graph having h edges can also be viewed as prefixes of Dyck pathsof bounded height h , since they start at the bottom but may end anywhere above or onthe bottom line. Their number is known to be equal to that of grand Dyck paths, of thesame length, which start in the middle of the band, may go above or below that line – aslong as they stay within bounds, and which we allow to end up either in the middle or justabove [5]. So each of the above sequences also counts constrained grand Dyck paths.More generally, walks can start anywhere in H (0 ≤ i ≤ h ), with the position along theroute always staying within the range [ ∶ h ] . Table 1 lists values for the number of paths Compiled already by Jonathon Bryant [1]. Usually, grand Dyck paths are defined to be of even length and to end up back on the starting line. To bemore inclusive, we allow odd-length grand Dyck paths that terminate one line above – as in [6], for instance– adopting the same moniker in the odd case, too. Accordingly, we can say that the number of grand Dyckpaths for 2 n and even h (with the up-down symmetry of the corridor) is always twice that for 2 n −
1. Seethe middle case of Table 1. The term “grand Dyck” is used in [15], for example; these lattice paths are alsoreferred to as “two-sided” or “bilateral” paths (e.g. [11]) on account of their shape, as “binomial” or “centralbinomial” paths (e.g. [13]) on account of their number, and as free
Dyck paths (e.g. [3]); they are classifiedas “bridges” in [2]. hrough a corridor of height h =
4, with one subtable for each starting point ( i = , , , , h =
5. These may be viewed as constrained versions of Pascal’s triangle,with each entry the sum of two prior entries. (Cf. [1].)In addition to pointing to the formula enumerating these more general sets of corridorpaths ending in an arbitrary window, we explore a beautiful symmetry between such sets ofpaths, those starting at level i and ending in a window of size 2 j + j and ending in a window of size 2 i + Main results
We use the notation J k ± j K as shorthand for a range [ k − j ∶ k + j + ] , which we make ofeven size, viz. 2 j +
2, by stretching the upper end one spot, to include k + j +
1. Thus, thewindow J k ± j K covers j + k .Our main result is the following intriguing equivalence: Theorem 1.
For all n, h ∈ N , k ∈ [ ∶ h ] , i, j ∈ [ ∶ min { k + , h − k }] : i n ⟿ h J k ± j K = j n ⟿ h J k ± i K (1)For example, 2 ⟿ J ± K = = ⟿ J ± K ; see Table 1. The bounds on i and j ensurethat the starting points are in H = [ ∶ h ] and that the target windows J k ± j K and J k ± i K do not extend beyond one row above or below the corridor H .Were i or j too big, k ± i or k ± j could extend too far beyond H , and the equality wouldnot hold, as is the case for 2 ⟿ J ± K = ≠ ⟿ J ± K = k ≤ h /
2, thetheorem holds as long as i, j ≤ k .The largest i and j can be (without being equal) is i = ⌊ h / ⌋ and j = ⌈ h / ⌉ , which gives ⌊ h / ⌋ n ⟿ H = ⌈ h / ⌉ n ⟿ H (2)and is no surprise.This theorem also holds for the degenerate case k = − i = j =
0, in which case the equivalence is true trivially.By up-down symmetry:
Lemma 2.
For all n, h ∈ N , i, j, k ∈ [ ∶ h ] , i n ⟿ h J k ± j K = ( h − i ) n ⟿ h J h − k − ± j K (3)So, for instances when i > h ÷
2, we can combine this lemma with our theorem to obtain:
Corollary 3.
The equivalence i n ⟿ h J k ± j K = j n ⟿ h J h − k − ± h − i K holds for all n, h ∈ N , k ∈ [ ∶ h ] , i ∈ [ max { k, h − k − } ∶ h ] , j ∈ [ ∶ min { k + , h − k }] . Lastly, the closed-form formula for the paths of interest is as follows: heorem 4. The number of corridor paths i n ⟿ h J k ± j K is ⌊ n / ⌋ ∑ z = ⌊ − n / ⌋ j ∑ s = ≤ k − j + s ≤ h [( n ⌈ n + k − i − j ⌉ + z ( h + ) + s ) − ( n ⌈ n + k + i − j ⌉ + z ( h + ) + s + )] for all n, h, j, k ∈ N , i ∈ [ ∶ h ] . Inductive proof
One can prove Theorem 1, viz. i n ⟿ h J k ± j K = j n ⟿ h J k ± i K by induction on the number of steps n , and with height h fixed throughout.Recall that the bounds on i and j are i, j ≥ i, j ≤ k + i, j ≤ h − k (6)The cases where either is out of bounds are excluded from the theorem.For n =
0, the starting and ending points must be the same. The two boundary conditions,viz. i ⟿ J k ± j K = [ k − j ≤ i ≤ k + j + ] j ⟿ J k ± i K = [ k − i ≤ j ≤ k + i + ] are equivalent since we are given that 0 ≤ i, j ≤ k + n > i n ⟿ J k ± j K = ( i − n − ⟿ J k ± j K ) + ( i + n − ⟿ J k ± j K ) basic recurrence = ( j n − ⟿ J k ± i − K ) + ( j n − ⟿ J k ± i + K ) induction = ( j n − ⟿ J k − i − ± K ) + ( j n − ⟿ J k ± i − K ) + ( j n − ⟿ J k + i + ± K ) definition = ( j n − ⟿ J k − ± i K ) + ( j n − ⟿ J k + ± i K ) definition = j n ⟿ J k ± i K basic recurrenceBut this only works if the two inductive cases also satisfy the theorem’s constraints.The problematic cases, when the inductive hypothesis cannot be applied, are three:(a) i =
0, since then i − < i − n − ⟿ J k ± j K ;(b) i = k +
1, since then i + > k + i + n − ⟿ J k ± j K ;(c) i = h − k , violating (6) for the right case. ortuitously, the exact same argument may be applied in the opposite direction, with therˆoles of i and j exchanged, to prove the identical equivalence: i n ⟿ J k ± j K = ( i n − ⟿ J k ± j − K ) + ( i n − ⟿ J k ± j + K ) (7) = ( j − n − ⟿ J k ± i K ) + ( j + n − ⟿ J k ± i K ) = j n ⟿ J k ± i K The cases for which this version of the argument is problematic are analogous but different:(a’) j = j = k +
1; or(c’) j = h − k .For the first exception (a), when i =
0, all is well with just one induction:0 n ⟿ J k ± j K = n − ⟿ J k ± j K = j n − ⟿ J k ± K = j n ⟿ J k ± K In the extreme case that k = h , and the induction is invalid, it must also be that j =
0, andthe equivalence holds immediately, sans induction. By the same token, case (a’) is also notan issue.Furthermore, whenever i = j , the theorem holds trivially, so the two combined cases (b,b’),when i = j = k +
1, and (c,c’), when i = j = h − k , are fine, too.So we only lack a proof for the following two combinations of the exceptions: (b,c’), when i = k + j = h − k , and (c,b’), when j = k + i = h − k . These are symmetric, so let’sdelve just into the second. Taking constraints (5,6) into account, we find that h = k + i = k = ⌊ h / ⌋ , and j = ⌈ h / ⌉ . So all we have to establish is the case ⌊ h / ⌋ n ⟿ H = ⌈ h / ⌉ n ⟿ H ,which we’ve already seen (2). 4. Combinatorial proof
One can derive the enumeration of Theorem 4 using a standard result for bounded latticepaths. Our main theorem will then follow as a corollary.The number M ( a, b, s, t ) of “monotonic” paths from ⟨ , ⟩ to ⟨ a, b ⟩ , taking a steps to theeast (E, → ) and b steps to the north (N, ↑ ), while totally avoiding (not touching or crossing)the boundaries y = x + s and y = x − t ( s, t ∈ Z + , t < b − a < s ) is known (by a reflectionargument) [8, 12, p. 6] to be M ( a, b, s, t ) = ∑ z ∈ Z [( a + bb + z ( s + t )) − ( a + bb + z ( s + t ) + t )] (8)with the (nonstandard) convention that ( nm ) = m ∉ N . See Figure 4.There is a straightforward relationship between these constrained N/E paths ⟨ , ⟩ ↝ ⟨ a, b ⟩ and those NE/SE paths ⟨ , i ⟩ ↝ ⟨ n, ℓ ⟩ that we have set out to study (as illustrated inFigure 2): n = a + b ℓ − i = b − at = i + s + t = h + lugging the solution a = n + i − ℓ b = n − i + ℓ s = h − i + t = i + i n ⟿ h ℓ = ∑ z ∈ Z [( n n − i + ℓ + z ( h + )) − ( n n − i + ℓ + z ( h + ) + i + )] (9)as long as 0 ≤ i, ℓ ≤ h . For those ℓ for which n − i + ℓ is not a whole number, the binomialcoefficients are all 0.Letting ℓ move along the window from k − j to k + j +
1, we get from (9) that i n ⟿ h J k ± j K = min { k + j + ,h } ∑ ℓ = max { ,k − j } ∑ z ∈ Z [( n n − i + ℓ + z ( h + )) − ( n n − i + ℓ + z ( h + ) + i + )] The sum for z can be restricted to the range ⌊ − n / ⌋ ∶ ⌊ n / ⌋ . Skipping over the impossibleodd or even values (for which the denominators of the binomial coefficients are fractional),we arrive at the stated formula of Theorem 4: i n ⟿ h J k ± j K = (10) ⌊ n / ⌋ ∑ z = ⌊ − n / ⌋ j ∑ s = ≤ k − j + s ≤ h [( n ⌈ n − i + k − j ⌉ + z ( h + ) + s ) − ( n ⌈ n + i + k − j ⌉ + z ( h + ) + s + )] Consider now only the cases considered in Theorem 1, which guarantee that k − j ≥ − k + j ≤ h +
1, so s may run from 0 to j without exception – bearing in mind (asshown above) that any instances when k − j + s = , h + s with j − s , we get i n ⟿ h J k ± j K = ∑ z ∈ Z j ∑ s = [( n ⌈ n − i + k − j ⌉ + z ( h + ) + s ) − ( n ⌈ n + i + k + j ⌉ + z ( h + ) − s + )] When j > i , the inner sums overlap (for s > i ) and cancel each other. So the above sum isalways equal to ∑ z ∈ Z [ min { i,j } ∑ s = ( nr + z ( h + ) + s ) − ( nr + z ( h + ) + i + j − s + )] where r = ⌈( n + k − i − j )/ ⌉ . This is symmetric in i and j ; hence Theorem 1. h = k = i = j = ℓ } ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ A T A T A T T T T A T AT T A T A T T A T A T A
Figure 5.
The 12-step blue path from level i = ℓ = ⟿ J ± K . Since k =
3, steps are labeled T whenthey start out towards the “attractor” y = . A when they head away in the opposite direction. So the blue path is labeled ATATATTTTATA . The target window size is j =
2, so we are in the ( < ) caseof the bijection. After seven (solid blue) steps ATATATT , the path touches y =
2, so the remaining 5 (dotted blue) steps,
TTATA , are copied as is andplaced with ⟨ , ⟩ as their initial point (dotted purple) , followed by the sevenin reverse (solid purple), that is, TTATATA , to obtain the corresponding path.The result is
TTATATTATATA , one of those counted by 2 ⟿ J ± K , whichall start from j = [ ∶ ] . Because of the unusual encoding, thereversed (solid) path segments do not actually resemble each other visually.The counterpart of the latter (purple) path is again the former (in blue), and isobtained by proceeding from the end towards the beginning until the windowsize becomes 2, per case ( > ). See Section 5 for details.5. Bijective proof
A bijection can be inferred from the inductive proof of Section 3 for the equivalence of theenumerations: i n ⟿ h J k ± j K = j n ⟿ h J k ± i K We use a novel representation for paths, which simplifies matters greatly.Draw a line y = k + /
2. Each step starting out towards that line is labeled T ; each headingaway is labeled A . From any given point, exactly one outgoing step ( ↗ or ↘ ) will be T andone A . We call this the TA representation of a lattice path (relative to k ). See Figure 5.Suppose the height of a point along the path is in the window [ k − j ∶ k + j + ] . If we takean A step from there, then the next point is in the wider window [ k − j − ∶ k + j + ] ; so j hasbeen incremented. Conversely, a T step brings it into the narrower range [ k − j + ∶ k + j ] ,with decremented j . Naturally, going backwards alongs the path has the opposite effect. f we take this point of view and go through the cases of the inductive proof, we find thatthe correspondence simply reverses the order of steps, either moving the last step to thebeginning or vice versa. When i = j , there is no need to do anything, since the two sides ofthe equivalence are identical. We are led to the following bijection between a path P startingat y = i and ending in the range [ k − j ∶ k + j + ] and its counterpart path P ∗ starting at y = j and ending in the range [ k − i ∶ k + i + ] :( = ) If i = j , then P ∗ = P .( < ) If i < j , follow the path from the start at level i until it reaches j , if ever. At thatpoint, we have P = QR , where y = j first transpires at the end of prefix Q . Then P ∗ = R ⟵ Q , where ⟵ Q is the reverse sequence of Q in its TA representation. If level j isnever attained, then R is empty, and P ∗ = ⟵ Q .( > ) If i > j , follow the path from the end backwards, starting with a target window ofsize j , moving leftwards until it grows to be i , if ever. A T step enlarges the window,while A shrinks it. If R is shortest suffix such that the window size is i at its onset,so that we have P = QR , then we let P ∗ = ⟵ RQ .It is not hard to verify that the transpositions involved keep the path within the boundedcorridor, given that the original path satisfied 0 ≤ i, j ≤ min { k + , h − k } .The path in Figure 2 is its own counterpart, as this is an instance of case ( = ) with i = j = Historical discussion
Theorem 1, our main result, is a significant generalization of the equality due to JohannCigler [5], namely, 0 n ⟿ h [ ∶ h ] = ̷ h n ⟿ h [ ̷ h ∶ ̷ h + ] (11)for all heights h , where ̷ h = h ÷ ̷ h n ⟿ [ ̷ h ∶ ̷ h + ] start inthe middle of the swath and end either in the middle – when the number of steps is even,or just above – when odd. As noted earlier, these are called “grand Dyck” paths. Dyckpath prefixes 0 n ⟿ [ ∶ h ] start at the bottom and end anywhere within the swath. Cigler’s(11) asserts the equality of cardinality of these two sets of paths. As such, it is a particularinstance of our more general result (1) with i = j = k = ̷ h . Phrased in our notation,Cigler proved: 0 n ⟿ h J ̷ h ± ̷ h K = ̷ h n ⟿ h J ̷ h ± K Cigler solicited alternative proofs of his result. More specifically, he asked in [4] for a bijective proof of the height h = n ⟿ [ ∶ ] = n ⟿ [ ∶ ] which gives rise to the Fibonacci numbers. The wished-for bijective solution to this veryparticular case was discovered shortly thereafter by Thomas Prellberg [4, Answer], followedby another due to Helmut Prodinger [14]. Most recently, Nancy Gu and Prodinger [10] onstructed a bijection for Cigler’s full case (11) by extending the idea in [14]. When thereare no upper and lower bounds on paths, there are long-standing well-known bijectionsbetween grand Dyck paths and Dyck path prefixes [7, 9].The bijection of the previous section supplies an alternative proof of Cigler’s (11). Inthat special case, the bijection amounts to simply reversing the order of steps in the TA representation. This works as is for even n in the grand Dyck ( i = k = ̷ h and j =
0) toDick-prefix ( i = j = k = ̷ h ) case of Cigler, as this is the ( i > j ) case of the bijectionand the window never grows too big (it may get to be ̷ h , the maximum excess of T movesover A moves, but no larger) to continue all way the beginning. Unfortunately, it doesn’tdo the trick when n is odd and h is even (because proceeding only backwards can lead to awindow wider than ̷ h = i ). For the odd n case, it is possible to modify the bijection by firstreversing the grand Dyck path left to right (so it ends on y = ̷ h but begins at i = ̷ h + TA representation and reversing. This now covers all cases of (11).The second bijection also works for even h and even n . For odd h , regardless of the parity of n , the first bijection actually succeeds for all i, j meeting the requirements of the theorem.So, when n and h have the same parity, both bijections work. In the more general cases,when k ≠ ̷ h , neither applies, and we resort to the slightly more complicated bijection of theprevious section, wherein only part of the TA path is reversed.We began our investigation seeking a bijective proof of (11). The simple bijection employ-ing the TA path encoding didn’t work in all cases. This led us to a sequence of generalizations,commencing from Cigler’s (11): 0 n ⟿ [ ∶ h ] = ̷ h n ⟿ [ ̷ h ∶ ̷ h + ] i n ⟿ [ ∶ h ] = ̷ h n ⟿ [ ̷ h − i ∶ ̷ h + i + ] i n ⟿ [ ̷ h − j ∶ ̷ h + j + ] = j n ⟿ [ ̷ h − i ∶ ̷ h + i + ] i n ⟿ [ k − j ∶ k + j + ] = j n ⟿ [ k − i ∶ k + i + ] First we let i be anywhere (not just 0), then we let j be any size (not just ̷ h ), and finallyallowed it to be centered at any k (not just ̷ h ). Concurrently, we programmed variousenumerations and potential bijections to lend support to – or refute – conjectures as theyarose. Casting the equivalence in a fashion that highlights its symmetry also contributed tofinding the generalizations and proofs.All the above variants share the basic idea that, as the starting point of one set of pathsmoves from the edge of the corridor towards the middle, the target range of the correspondingequinumerous set of paths grows wider and wider. This behavior is what suggested the TA encoding in the first place. Acknowledgment
I gratefully thank Johann Cigler for both encouragement and references and ChristianRinderknecht for first bringing Cigler’s interesting challenge to my attention.
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CongressusNumerantium , 143:193–205, December 2000.[16] Neil J. A. Sloane et al. The on-line encyclopedia of integer sequences, 2020. http://oeis.org . n = ℓ OEIS4 1 1 2 5 14 41 122 365 1094 4 A0070513 1 2 5 14 41 122 365 1094 3 A0070512 1 3 9 27 81 243 729 2187 2 A0002441 1 4 13 40 121 364 1093 1 A0034620 1 4 13 40 121 364 1093 0 A0034624 1 2 5 14 41 122 365 1094 4 A0070513 1 2 5 14 41 122 365 1094 3281 3 A0070512 1 3 9 27 81 243 729 2187 2 A0002441 1 4 13 40 121 364 1093 3280 1 A0034620 1 4 13 40 121 364 1093 0 A0034624 1 3 9 27 81 243 729 2187 4 A0002443 1 3 9 27 81 243 729 2187 3 A0002442 1 2 6 18 54 162 486 1458 4374 2 A0251921 1 3 9 27 81 243 729 2187 1 A0002440 1 3 9 27 81 243 729 2187 0 A0002444 1
13 40 121 364 1093 4 A0034623
40 121
729 2187 2 A0002441
41 122
365 1094 3281 1 A0070510 1 2 5 14
122 365 1094 0 A0070514 1 4 13 40 121 364 1093 4 A0034623 1 4 13 40 121 364 1093 3 A0034622 1 3 9 27 81 243 729 2187 2 A0002441 1 2 5 14 41 122 365 1094 1 A0070510 1 1 2 5 14 41 122 365 1094 0 A007051
Table 1.
The number of paths i n ⟿ ℓ for i, ℓ ∈ [ ∶ ] , n ∈ [ ∶ ] . Forexample, 2 ⟿ = ⟿ [ ∶ ] = ⟿ = ⟿ [ ∶ ] = blueboldface . Sloane numbers of the sequences are provided in the last column. n = ℓ OEIS5 1 1 2 5 14 42 131 417 1341 5 A0809374 1 2 5 14 42 131 417 1341 4 A0809373 1 3 9 28 89 286 924 2993 3 A0947902 1 4 14 47 155 507 1652 2 A0060531 1 5 19 66 221 728 2380 1 A0050210 1 5 19 66 221 728 0 A0050215 1 2 5 14 42 131 417 1341 5 A0809374 1 2 5 14 42 131 417 1341 4334 4 A0809373 1 3 9 28 89 286 924 2993 3 A0947902 1 4 14 47 155 507 1652 5373 2 A0060531 1 5 19 66 221 728 2380 1 A0050210 1 5 19 66 221 728 2380 0 A0050215 1 3 9 28 89 286 924 2993 5 A0947904 1 3 9 28 89 286 924 2993 4 A0947903 1 2 6 19 61 197 638 2069 6714 3 A0529752 1 3 10 33 108 352 1145 3721 2 A0605571 1 4 14 47 155 507 1652 5373 1 A0060530 1 4 14 47 155 507 1652 0 A0060535 1 4 14 47 155 507 1652 5 A0060534 1 4 14 47 155 507 1652 5373 4 A0060533 1 3 10 33 108 352 1145 3721 3 A0605572 1 2 6 19 61 197 638 2069 6714 2 A0529751 1 3 9 28 89 286 924 2993 1 A0947900 1 3 9 28 89 286 924 2993 0 A0947905 1 5 19 66 221 728 2380 5 A0050214 1 5 19 66 221 728 2380 4 A0050213 1 4 14 47 155 507 1652 5373 3 A0060532 1 3 9 28 89 286 924 2993 2 A0947901 1 2 5 14 42 131 417 1341 4334 1 A0809370 1 2 5 14 42 131 417 1341 0 A0809375 1 5 19 66 221 728 5 A0050214 1 5 19 66 221 728 2380 4 A0050213 1 4 14 47 155 507 1652 3 A0060532 1 3 9 28 89 286 924 2993 2 A0947901 1 2 5 14 42 131 417 1341 1 A0809370 1 1 2 5 14 42 131 417 1341 0 A080937
Table 2.
Paths i n ⟿ ℓ constrained to height 5, n ∈ [ ∶ ] . Sloane numbersof the sequences are provided in the last column.. Sloane numbersof the sequences are provided in the last column.