Bicolored Motzkin Paths in Combinatorics
Updated 30 November 2025
Bicolored Motzkin paths are lattice paths from (0,0) to (n,0) that use up, down, and horizontal steps with two distinct colors and nested parenthetical constraints.
They are analyzed via recurrence relations and generating functions, providing explicit formulas and bijections to combinatorial objects like Young tableaux and interval-closed sets.
Their rich combinatorial structure supports applications in lattice path enumeration, poset theory, and algebraic combinatorics.
A bicolored Motzkin path (also “2-Motzkin path” or “two-colored Motzkin path”) is a lattice path from ( 0 , 0 ) (0,0) ( 0 , 0 ) ( n , 0 ) (n,0) ( n , 0 ) x x x Motzkin paths by introducing color assignments to step types and are a central tool for establishing bijections in algebraic combinatorics and enumerative geometry, notably with interval-closed sets, Young tableaux of bounded height, and constrained lattice walks. Their paper encompasses precise recursions, generating functions, closed formulae, and complex bijections to combinatorial structures.
1. Combinatorial Definitions
A bicolored Motzkin path of length n n n s 1 s 2 ⋯ s n s_1s_2\cdots s_n s 1 s 2 ⋯ s n ( x , y ) (x,y) ( x , y ) ( x + 1 , y ′ ) (x+1,y') ( x + 1 , y ′ ) y ′ ∈ { y − 1 , y , y + 1 } y'\in\{y-1,\,y,\,y+1\} y ′ ∈ { y − 1 , y , y + 1 } y y y
General colored (“2-Motzkin”) case (Eu et al., 2013 ):
Steps:
U ( 1 ) = ( 1 , 1 ) U^{(1)}=(1,1) U ( 1 ) = ( 1 , 1 ) U ( 2 ) = ( 1 , 1 ) U^{(2)}=(1,1) U ( 2 ) = ( 1 , 1 ) D ( 1 ) = ( 1 , − 1 ) D^{(1)}=(1,-1) D ( 1 ) = ( 1 , − 1 ) D ( 2 ) = ( 1 , − 1 ) D^{(2)}=(1,-1) D ( 2 ) = ( 1 , − 1 ) L = ( 1 , 0 ) L=(1,0) L = ( 1 , 0 ) All subwords for each k = 1 , 2 k=1,2 k = 1 , 2 U ( k ) , D ( k ) U^{(k)},D^{(k)} U ( k ) , D ( k )
Every ( U ( 2 ) , D ( 2 ) ) (U^{(2)},D^{(2)}) ( U ( 2 ) , D ( 2 ) ) ( U ( 1 ) , D ( 1 ) ) (U^{(1)},D^{(1)}) ( U ( 1 ) , D ( 1 ) )
For every prefix μ \mu μ
# U ( 1 ) ( μ ) − # D ( 1 ) ( μ ) ≥ # U ( 2 ) ( μ ) − # D ( 2 ) ( μ ) ≥ 0 \#U^{(1)}(\mu)-\#D^{(1)}(\mu)\ge\#U^{(2)}(\mu)-\#D^{(2)}(\mu)\ge 0 # U ( 1 ) ( μ ) − # D ( 1 ) ( μ ) ≥ # U ( 2 ) ( μ ) − # D ( 2 ) ( μ ) ≥ 0
Basic bicolored case (Krattenthaler, 15 Jan 2025 ):
Steps:
U = ( 1 , 1 ) U=(1,1) U = ( 1 , 1 ) D = ( 1 , − 1 ) D=(1,-1) D = ( 1 , − 1 ) H U = ( 1 , 0 ) H_U=(1,0) H U = ( 1 , 0 ) H D = ( 1 , 0 ) H_D=(1,0) H D = ( 1 , 0 ) No step ever below y = 0 y=0 y = 0
u ( P ) = d ( P ) u(P)=d(P) u ( P ) = d ( P ) y = 0 y=0 y = 0 Interval-closed set version (Elizalde et al., 20 Dec 2024 ):
Steps: U ( 1 , 1 ) U\ (1,1) U ( 1 , 1 ) D ( 1 , − 1 ) D\ (1,-1) D ( 1 , − 1 ) H 1 ( 1 , 0 ) H_1\ (1,0) H 1 ( 1 , 0 ) H 2 ( 1 , 0 ) H_2\ (1,0) H 2 ( 1 , 0 )
Bijection constraints (see Section 2).
This diversity reflects the role of coloring—either as a local decoration or tied to more stringent parenthetical constraints.
2. Recurrence Relations and Generating Functions
Bicolored Motzkin paths admit rich recursive structure and generating functions.
M n ( 2 ) = M n − 1 ( 2 ) + 2 ∑ i = 0 n − 2 M i ( 2 ) M n − 2 − i ( 2 ) , M 0 ( 2 ) = 1 M_n^{(2)} = M_{n-1}^{(2)} + 2\sum_{i=0}^{n-2} M_i^{(2)} M_{n-2-i}^{(2)}, \qquad M_0^{(2)} = 1 M n ( 2 ) = M n − 1 ( 2 ) + 2 i = 0 ∑ n − 2 M i ( 2 ) M n − 2 − i ( 2 ) , M 0 ( 2 ) = 1
Functional equation:
M ( 2 ) ( x ) = 1 + x M ( 2 ) ( x ) + 2 x 2 ( M ( 2 ) ( x ) ) 2 M^{(2)}(x) = 1 + x M^{(2)}(x) + 2x^2\left(M^{(2)}(x)\right)^2 M ( 2 ) ( x ) = 1 + x M ( 2 ) ( x ) + 2 x 2 ( M ( 2 ) ( x ) ) 2
Closed form (unique solution vanishing at x = 0 x=0 x = 0
M ( 2 ) ( x ) = 1 − x − ( 1 − x ) 2 − 8 x 2 4 x 2 M^{(2)}(x) = \frac{1-x-\sqrt{(1-x)^2-8x^2}}{4x^2} M ( 2 ) ( x ) = 4 x 2 1 − x − ( 1 − x ) 2 − 8 x 2
Explicit coefficients:
M n ( 2 ) = ∑ j = 0 ⌊ n / 2 ⌋ 2 j j + 1 ( 2 j j ) ( n 2 j ) M_n^{(2)} = \sum_{j=0}^{\lfloor n/2\rfloor} \frac{2^j}{j+1} \binom{2j}{j} \binom{n}{2j} M n ( 2 ) = j = 0 ∑ ⌊ n /2 ⌋ j + 1 2 j ( j 2 j ) ( 2 j n )
M ( x ) = 1 + 2 x M ( x ) + x 2 M 2 ( x ) M(x) = 1 + 2x M(x) + x^2 M^2(x) M ( x ) = 1 + 2 x M ( x ) + x 2 M 2 ( x )
M ( x ) = 1 − 2 x − 1 − 4 x 2 x 2 M(x) = \frac{1-2x-\sqrt{1-4x}}{2x^2} M ( x ) = 2 x 2 1 − 2 x − 1 − 4 x
For statistics of up/down and horizontal steps,
M ( x , y , z ) = 1 + 2 x z M ( x , y , z ) + x 2 y 2 M 2 ( x , y , z ) M(x, y, z) = 1 + 2xz M(x, y, z) + x^2 y^2 M^2(x, y, z) M ( x , y , z ) = 1 + 2 x z M ( x , y , z ) + x 2 y 2 M 2 ( x , y , z )
Leading to coefficient extraction via Lagrange inversion:
[ x n y 2 q z j ] M ( x , y , z ) = n ! j ! q ! ( q + 1 ) ! 2 j , with n = j + 2 q [x^n y^{2q} z^j] M(x,y,z) = \frac{n!}{j!\,q!\,(q+1)!} 2^j, ~~ \text{with}~ n = j + 2q [ x n y 2 q z j ] M ( x , y , z ) = j ! q ! ( q + 1 )! n ! 2 j , with n = j + 2 q
C ( x , y ) = 1 + ( x + y ) C ( x , y ) + x y C 2 ( x , y ) C(x,y) = 1 + (x+y)C(x,y) + x y C^2(x,y) C ( x , y ) = 1 + ( x + y ) C ( x , y ) + x y C 2 ( x , y )
C ( x , y ) = 1 − x − y − ( 1 − x − y ) 2 − 4 x y 2 x y C(x,y) = \frac{1-x-y-\sqrt{(1-x-y)^2 - 4xy}}{2xy} C ( x , y ) = 2 x y 1 − x − y − ( 1 − x − y ) 2 − 4 x y < / p > < p > < e m > A l l t h e s e r e c u r r e n c e s a r e v a r i a n t s o f t h e M o t z k i n – S c h r o ¨ d e r b a c k b o n e , t a i l o r e d w i t h c o l o r i n g a n d c o m b i n a t o r i a l c o n s t r a i n t s t o e n c o d e r i c h e r o b j e c t c l a s s e s . < / e m > < / p > < h 2 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ b i j e c t i o n s − t o − a l g e b r a i c − a n d − g e o m e t r i c − o b j e c t s ′ > 3. B i j e c t i o n s t o A l g e b r a i c a n d G e o m e t r i c O b j e c t s < / h 2 > < p > B i c o l o r e d M o t z k i n p a t h s e m e r g e n a t u r a l l y a s c a n o n i c a l r e p r e s e n t a t i v e s i n b i j e c t i o n s t o a v a r i e t y o f c o m b i n a t o r i a l o b j e c t s . < / p > < u l > < l i > < s t r o n g > S t a n d a r d Y o u n g t a b l e a u x ( S Y T ) o f b o u n d e d h e i g h t ( < a h r e f = " / p a p e r s / 1302.3012 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > E u e t a l . , 2013 < / a > ) : < / s t r o n g > < u l > < l i > T h e r e i s a b i j e c t i o n b e t w e e n 2 − M o t z k i n p a t h s o f l e n g t h </p>
<p><em>All these recurrences are variants of the Motzkin–Schröder backbone, tailored with coloring and combinatorial constraints to encode richer object classes.</em></p>
<h2 class='paper-heading' id='bijections-to-algebraic-and-geometric-objects'>3. Bijections to Algebraic and Geometric Objects</h2>
<p>Bicolored Motzkin paths emerge naturally as canonical representatives in bijections to a variety of combinatorial objects.</p>
<ul>
<li><strong>Standard Young tableaux (SYT) of bounded height (<a href="/papers/1302.3012" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Eu et al., 2013</a>):</strong>
<ul>
<li>There is a bijection between 2-Motzkin paths of length < / p >< p >< e m > A llt h eserec u rre n ces a re v a r ian t so f t h e M o t z kin – S c h r o ¨ d er ba c kb o n e , t ai l ore d w i t h co l or in g an d co mbina t or ia l co n s t r ain t s t oe n co d er i c h ero bj ec t c l a sses . < / e m >< / p >< h 2 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ bij ec t i o n s − t o − a l g e b r ai c − an d − g eo m e t r i c − o bj ec t s ′ > 3. B ij ec t i o n s t o A l g e b r ai c an d G eo m e t r i c O bj ec t s < / h 2 >< p > B i co l ore d M o t z kin p a t h se m er g e na t u r a ll y a sc an o ni c a l re p rese n t a t i v es inbij ec t i o n s t o a v a r i e t yo f co mbina t or ia l o bj ec t s . < / p >< u l >< l i >< s t ro n g > St an d a r d Y o u n g t ab l e a ux ( S Y T ) o f b o u n d e d h e i g h t ( < ah re f = "/ p a p ers /1302.3012" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > E u e t a l . , 2013 < / a > ) :< / s t ro n g >< u l >< l i > T h ere i s abij ec t i o nb e tw ee n 2 − M o t z kin p a t h so f l e n g t h a n d S Y T o f s i z e and SYT of size an d S Y T o f s i ze a n d a t m o s t and at most an d a t m os t r o w s . < / l i > < l i > T h e b i j e c t i o n p r o c e e d s b y r e c u r s i v e l y “ p e e l i n g o f f ” f i r s t − r o w t a b l e a u e n t r i e s a n d m a t c h i n g w i t h l e v e l o r u p – d o w n p a i r s , t r a c k i n g c o l o r t o e n f o r c e n e s t e d s t r u c t u r e . < / l i > < l i > T h e n u m b e r o f u n c o l o r e d l e v e l s t e p s i n t h e b i c o l o r e d M o t z k i n p a t h e q u a l s t h e n u m b e r o f o d d − l e n g t h c o l u m n s i n t h e t a b l e a u . < / l i > < / u l > < / l i > < l i > < s t r o n g > I n t e r v a l − c l o s e d s e t s i n t h e p o s e t rows.</li>
<li>The bijection proceeds by recursively “peeling off” first-row tableau entries and matching with level or up–down pairs, tracking color to enforce nested structure.</li>
<li>The number of uncolored level steps in the bicolored Motzkin path equals the number of odd-length columns in the tableau.</li>
</ul></li>
<li><strong>Interval-closed sets in the poset ro w s . < / l i >< l i > T h e bij ec t i o n p rocee d s b yrec u rs i v e l y “ p ee l in g o ff ” f i rs t − ro wt ab l e a u e n t r i es an d ma t c hin g w i t h l e v e l or u p – d o w n p ai rs , t r a c kin g co l or t oe n f orce n es t e d s t r u c t u re . < / l i >< l i > T h e n u mb ero f u n co l ore d l e v e l s t e p s in t h e bi co l ore d M o t z kin p a t h e q u a l s t h e n u mb ero f o dd − l e n g t h co l u mn s in t h e t ab l e a u . < / l i >< / u l >< / l i >< l i >< s t ro n g > I n t er v a l − c l ose d se t s in t h e p ose t ( < a h r e f = " / p a p e r s / 2412.16368 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > E l i z a l d e e t a l . , 20 D e c 2024 < / a > ) : < / s t r o n g > < u l > < l i > E v e r y i n t e r v a l − c l o s e d s e t i s i n b i j e c t i o n w i t h a b i c o l o r e d M o t z k i n p a t h s a t i s f y i n g : < / l i > < l i > (<a href="/papers/2412.16368" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Elizalde et al., 20 Dec 2024</a>):</strong>
<ul>
<li>Every interval-closed set is in bijection with a bicolored Motzkin path satisfying:</li>
<li> ( < ah re f = "/ p a p ers /2412.16368" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > El i z a l d ee t a l . , 20 Dec 2024 < / a > ) :< / s t ro n g >< u l >< l i > E v ery in t er v a l − c l ose d se t i s inbij ec t i o n w i t habi co l ore d M o t z kin p a t h s a t i s f y in g :< / l i >< l i > < / l i > < l i > </li>
<li> < / l i >< l i > < / l i > < l i > N o </li>
<li>No < / l i >< l i > N o s t e p o n t h e step on the s t e p o n t h e − a x i s i s i m m e d i a t e l y f o l l o w e d b y -axis is immediately followed by − a x i s i s imm e d ia t e l y f o ll o w e d b y . < / l i > < l i > G e n e r a t i n g f u n c t i o n .</li>
<li>Generating function . < / l i >< l i > G e n er a t in g f u n c t i o n e n u m e r a t e s a l l s u c h s e t s v i a t h e s e p a t h s . < / l i > < / u l > < / l i > < l i > < s t r o n g > S e t − v a l u e d s t a n d a r d t a b l e a u x ( < a h r e f = " / p a p e r s / 2501.09208 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > K r a t t e n t h a l e r , 15 J a n 2025 < / a > ) : < / s t r o n g > < u l > < l i > T w o − r o w e d s e t − v a l u e d s t a n d a r d t a b l e a u x a r e i n b i j e c t i o n w i t h b i c o l o r e d M o t z k i n p a t h s . < / l i > < l i > E a c h e n t r y i n t h e f i r s t r o w c o r r e s p o n d s t o a n u p − s t e p , i n t h e s e c o n d r o w t o a d o w n − s t e p , a n d a d d i t i o n a l e n t r i e s t o c o l o r e d h o r i z o n t a l s . < / l i > < / u l > < / l i > < l i > < s t r o n g > Q u a r t e r − p l a n e a n d t r i a n g u l a r w a l k s ( < a h r e f = " / p a p e r s / 1412.1504 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > Y e a t s , 2014 < / a > , < a h r e f = " / p a p e r s / 2007.08868 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > C o u r t i e l e t a l . , 2020 < / a > ) : < / s t r o n g > < u l > < l i > Q u a r t e r − p l a n e w a l k s w i t h s p e c i f i e d s t e p s m a p b i j e c t i v e l y t o b i c o l o r e d M o t z k i n p a t h s . < / l i > < l i > T r i a n g u l a r w a l k s i n s i x d i r e c t i o n s ( r e s t r i c t e d d o m a i n ) m a t c h t w o − c o l o r e d M o t z k i n p a t h s o f b o u n d e d h e i g h t , w i t h h o r i z o n t a l s t e p s f o r b i d d e n a t m a x i m a l h e i g h t . < / l i > < l i > T h e “ s c a f f o l d i n g ” b i j e c t i o n ( < a h r e f = " / p a p e r s / 2007.08868 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > C o u r t i e l e t a l . , 2020 < / a > ) p r o v i d e s a l i n e a r − t i m e m a p p i n g a s s o c i a t i n g t h e c e l l w i s e p r o f i l e o f d o m a i n p o i n t s t o M o t z k i n p a t h s t e p s . < / l i > < / u l > < / l i > < / u l > < h 2 c l a s s = ′ p a p e r − h e a d i n g ′ i d = ′ e n u m e r a t i v e − c o n s e q u e n c e s − a n d − e x p l i c i t − f o r m u l a s ′ > 4. E n u m e r a t i v e C o n s e q u e n c e s a n d E x p l i c i t F o r m u l a s < / h 2 > < p > E n u m e r a t i o n f o r m u l a s a r e b o t h e x p l i c i t a n d r e f i n e d i n t e r m s o f s t e p s t a t i s t i c s . < / p > < u l > < l i > < s t r o n g > E x a m p l e v a l u e s f o r enumerates all such sets via these paths.</li>
</ul></li>
<li><strong>Set-valued standard tableaux (<a href="/papers/2501.09208" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Krattenthaler, 15 Jan 2025</a>):</strong>
<ul>
<li>Two-rowed set-valued standard tableaux are in bijection with bicolored Motzkin paths.</li>
<li>Each entry in the first row corresponds to an up-step, in the second row to a down-step, and additional entries to colored horizontals.</li>
</ul></li>
<li><strong>Quarter-plane and triangular walks (<a href="/papers/1412.1504" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Yeats, 2014</a>, <a href="/papers/2007.08868" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Courtiel et al., 2020</a>):</strong>
<ul>
<li>Quarter-plane walks with specified steps map bijectively to bicolored Motzkin paths.</li>
<li>Triangular walks in six directions (restricted domain) match two-colored Motzkin paths of bounded height, with horizontal steps forbidden at maximal height.</li>
<li>The “scaffolding” bijection (<a href="/papers/2007.08868" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Courtiel et al., 2020</a>) provides a linear-time mapping associating the cellwise profile of domain points to Motzkin path steps.</li>
</ul></li>
</ul>
<h2 class='paper-heading' id='enumerative-consequences-and-explicit-formulas'>4. Enumerative Consequences and Explicit Formulas</h2>
<p>Enumeration formulas are both explicit and refined in terms of step statistics.</p>
<ul>
<li><strong>Example values for e n u m er a t es a ll s u c h se t s v ia t h ese p a t h s . < / l i >< / u l >< / l i >< l i >< s t ro n g > S e t − v a l u e d s t an d a r d t ab l e a ux ( < ah re f = "/ p a p ers /2501.09208" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > Kr a tt e n t ha l er , 15 J an 2025 < / a > ) :< / s t ro n g >< u l >< l i > Tw o − ro w e d se t − v a l u e d s t an d a r d t ab l e a ux a re inbij ec t i o n w i t hbi co l ore d M o t z kin p a t h s . < / l i >< l i > E a c h e n t ry in t h e f i rs t ro w corres p o n d s t o an u p − s t e p , in t h eseco n d ro wt o a d o w n − s t e p , an d a dd i t i o na l e n t r i es t oco l ore d h or i zo n t a l s . < / l i >< / u l >< / l i >< l i >< s t ro n g > Q u a r t er − pl an e an d t r ian gu l a r w a l k s ( < ah re f = "/ p a p ers /1412.1504" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > Y e a t s , 2014 < / a > , < ah re f = "/ p a p ers /2007.08868" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > C o u r t i e l e t a l . , 2020 < / a > ) :< / s t ro n g >< u l >< l i > Q u a r t er − pl an e w a l k s w i t h s p ec i f i e d s t e p s ma p bij ec t i v e l y t o bi co l ore d M o t z kin p a t h s . < / l i >< l i > T r ian gu l a r w a l k s in s i x d i rec t i o n s ( res t r i c t e dd o main ) ma t c h tw o − co l ore d M o t z kin p a t h so f b o u n d e d h e i g h t , w i t hh or i zo n t a l s t e p s f or bi dd e na t ma x ima l h e i g h t . < / l i >< l i > T h e “ sc a ff o l d in g ” bij ec t i o n ( < ah re f = "/ p a p ers /2007.08868" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > C o u r t i e l e t a l . , 2020 < / a > ) p ro v i d es a l in e a r − t im e ma pp in g a ssoc ia t in g t h ece llw i se p ro f i l eo fd o main p o in t s t o M o t z kin p a t h s t e p s . < / l i >< / u l >< / l i >< / u l >< h 2 c l a ss = ′ p a p er − h e a d in g ′ i d = ′ e n u m er a t i v e − co n se q u e n ces − an d − e x pl i c i t − f or m u l a s ′ > 4. E n u m er a t i v e C o n se q u e n ces an d E x pl i c i tF or m u l a s < / h 2 >< p > E n u m er a t i o n f or m u l a s a re b o t h e x pl i c i t an d re f in e d in t er m so f s t e p s t a t i s t i cs . < / p >< u l >< l i >< s t ro n g > E x am pl e v a l u es f or ( < a h r e f = " / p a p e r s / 1302.3012 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > E u e t a l . , 2013 < / a > ) : < / s t r o n g > < / l i > < / u l > < d i v c l a s s = ′ o v e r f l o w − x − a u t o m a x − w − f u l l m y − 4 ′ > < t a b l e c l a s s = ′ t a b l e b o r d e r − c o l l a p s e w − f u l l ′ s t y l e = ′ t a b l e − l a y o u t : f i x e d ′ > < t h e a d > < t r > < t h > (<a href="/papers/1302.3012" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Eu et al., 2013</a>):</strong></li>
</ul>
<div class='overflow-x-auto max-w-full my-4'><table class='table border-collapse w-full' style='table-layout: fixed'><thead><tr>
<th> ( < ah re f = "/ p a p ers /1302.3012" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > E u e t a l . , 2013 < / a > ) :< / s t ro n g >< / l i >< / u l >< d i v c l a ss = ′ o v er f l o w − x − a u t o ma x − w − f u ll m y − 4 ′ >< t ab l ec l a ss = ′ t ab l e b or d er − co ll a p se w − f u l l ′ s t y l e = ′ t ab l e − l a yo u t : f i x e d ′ >< t h e a d >< t r >< t h > < / t h > < t h > </th>
<th> < / t h >< t h > < / t h > < / t r > < / t h e a d > < t b o d y > < t r > < t d > 0 < / t d > < t d > 1 < / t d > < / t r > < t r > < t d > 1 < / t d > < t d > 1 < / t d > < / t r > < t r > < t d > 2 < / t d > < t d > 3 < / t d > < / t r > < t r > < t d > 3 < / t d > < t d > 7 < / t d > < / t r > < t r > < t d > 4 < / t d > < t d > 21 < / t d > < / t r > < t r > < t d > 5 < / t d > < t d > 61 < / t d > < / t r > < / t b o d y > < / t a b l e > < / d i v > < u l > < l i > < s t r o n g > R e f i n e d m u l t i n o m i a l e n u m e r a t i o n ( < a h r e f = " / p a p e r s / 2501.09208 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > K r a t t e n t h a l e r , 15 J a n 2025 < / a > ) : < / s t r o n g > < / l i > < / u l > < p > F o r b i c o l o r e d M o t z k i n p a t h s o f l e n g t h </th>
</tr>
</thead><tbody><tr>
<td>0</td>
<td>1</td>
</tr>
<tr>
<td>1</td>
<td>1</td>
</tr>
<tr>
<td>2</td>
<td>3</td>
</tr>
<tr>
<td>3</td>
<td>7</td>
</tr>
<tr>
<td>4</td>
<td>21</td>
</tr>
<tr>
<td>5</td>
<td>61</td>
</tr>
</tbody></table></div>
<ul>
<li><strong>Refined multinomial enumeration (<a href="/papers/2501.09208" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Krattenthaler, 15 Jan 2025</a>):</strong></li>
</ul>
<p>For bicolored Motzkin paths of length < / t h >< / t r >< / t h e a d >< t b o d y >< t r >< t d > 0 < / t d >< t d > 1 < / t d >< / t r >< t r >< t d > 1 < / t d >< t d > 1 < / t d >< / t r >< t r >< t d > 2 < / t d >< t d > 3 < / t d >< / t r >< t r >< t d > 3 < / t d >< t d > 7 < / t d >< / t r >< t r >< t d > 4 < / t d >< t d > 21 < / t d >< / t r >< t r >< t d > 5 < / t d >< t d > 61 < / t d >< / t r >< / t b o d y >< / t ab l e >< / d i v >< u l >< l i >< s t ro n g > R e f in e d m u lt in o mia l e n u m er a t i o n ( < ah re f = "/ p a p ers /2501.09208" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > Kr a tt e n t ha l er , 15 J an 2025 < / a > ) :< / s t ro n g >< / l i >< / u l >< p > F or bi co l ore d M o t z kin p a t h so f l e n g t h w i t h with w i t h u p / d o w n a n d up/down and u p / d o w nan d h o r i z o n t a l s t e p s , < / p > < p > horizontal steps,</p>
<p> h or i zo n t a l s t e p s , < / p >< p > < / p > < p > w i t h </p>
<p>with < / p >< p > w i t h , , , , v a n i s h i n g f o r o d d , vanishing for odd , v ani s hin g f oro dd . < / p > < u l > < l i > < s t r o n g > I n t e r v a l − c l o s e d s e t s i n p r o d u c t o f c h a i n s ( < a h r e f = " / p a p e r s / 2412.16368 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > E l i z a l d e e t a l . , 20 D e c 2024 < / a > ) : < / s t r o n g > < / l i > < / u l > < d i v c l a s s = ′ o v e r f l o w − x − a u t o m a x − w − f u l l m y − 4 ′ > < t a b l e c l a s s = ′ t a b l e b o r d e r − c o l l a p s e w − f u l l ′ s t y l e = ′ t a b l e − l a y o u t : f i x e d ′ > < t h e a d > < t r > < t h > .</p>
<ul>
<li><strong>Interval-closed sets in product of chains (<a href="/papers/2412.16368" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Elizalde et al., 20 Dec 2024</a>):</strong></li>
</ul>
<div class='overflow-x-auto max-w-full my-4'><table class='table border-collapse w-full' style='table-layout: fixed'><thead><tr>
<th> . < / p >< u l >< l i >< s t ro n g > I n t er v a l − c l ose d se t s in p ro d u c t o f c hain s ( < ah re f = "/ p a p ers /2412.16368" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > El i z a l d ee t a l . , 20 Dec 2024 < / a > ) :< / s t ro n g >< / l i >< / u l >< d i v c l a ss = ′ o v er f l o w − x − a u t o ma x − w − f u ll m y − 4 ′ >< t ab l ec l a ss = ′ t ab l e b or d er − co ll a p se w − f u l l ′ s t y l e = ′ t ab l e − l a yo u t : f i x e d ′ >< t h e a d >< t r >< t h > < / t h > < t h > 0 < / t h > < t h > 1 < / t h > < t h > 2 < / t h > < t h > 3 < / t h > < / t r > < / t h e a d > < t b o d y > < t r > < t d > 0 < / t d > < t d > 1 < / t d > < t d > 1 < / t d > < t d > 1 < / t d > < t d > 1 < / t d > < / t r > < t r > < t d > 1 < / t d > < t d > 1 < / t d > < t d > 2 < / t d > < t d > 4 < / t d > < t d > 7 < / t d > < / t r > < t r > < t d > 2 < / t d > < t d > 1 < / t d > < t d > 4 < / t d > < t d > 13 < / t d > < t d > 33 < / t d > < / t r > < t r > < t d > 3 < / t d > < t d > 1 < / t d > < t d > 7 < / t d > < t d > 30 < / t d > < t d > 114 < / t d > < / t r > < / t b o d y > < / t a b l e > < / d i v > < u l > < l i > < s t r o n g > C a t a l a n r e p r e s e n t a t i o n ( < a h r e f = " / p a p e r s / 2012.14947 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > D e J a g e r e t a l . , 2020 < / a > < a h r e f = " / p a p e r s / 2501.09208 " t i t l e = " " r e l = " n o f o l l o w " d a t a − t u r b o = " f a l s e " c l a s s = " a s s i s t a n t − l i n k " x − d a t a x − t o o l t i p . r a w = " " > K r a t t e n t h a l e r , 15 J a n 2025 < / a > ) : < / s t r o n g > < / l i > < / u l > < p > </th>
<th>0</th>
<th>1</th>
<th>2</th>
<th>3</th>
</tr>
</thead><tbody><tr>
<td>0</td>
<td>1</td>
<td>1</td>
<td>1</td>
<td>1</td>
</tr>
<tr>
<td>1</td>
<td>1</td>
<td>2</td>
<td>4</td>
<td>7</td>
</tr>
<tr>
<td>2</td>
<td>1</td>
<td>4</td>
<td>13</td>
<td>33</td>
</tr>
<tr>
<td>3</td>
<td>1</td>
<td>7</td>
<td>30</td>
<td>114</td>
</tr>
</tbody></table></div>
<ul>
<li><strong>Catalan representation (<a href="/papers/2012.14947" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">DeJager et al., 2020</a><a href="/papers/2501.09208" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Krattenthaler, 15 Jan 2025</a>):</strong></li>
</ul>
<p> < / t h >< t h > 0 < / t h >< t h > 1 < / t h >< t h > 2 < / t h >< t h > 3 < / t h >< / t r >< / t h e a d >< t b o d y >< t r >< t d > 0 < / t d >< t d > 1 < / t d >< t d > 1 < / t d >< t d > 1 < / t d >< t d > 1 < / t d >< / t r >< t r >< t d > 1 < / t d >< t d > 1 < / t d >< t d > 2 < / t d >< t d > 4 < / t d >< t d > 7 < / t d >< / t r >< t r >< t d > 2 < / t d >< t d > 1 < / t d >< t d > 4 < / t d >< t d > 13 < / t d >< t d > 33 < / t d >< / t r >< t r >< t d > 3 < / t d >< t d > 1 < / t d >< t d > 7 < / t d >< t d > 30 < / t d >< t d > 114 < / t d >< / t r >< / t b o d y >< / t ab l e >< / d i v >< u l >< l i >< s t ro n g > C a t a l an re p rese n t a t i o n ( < ah re f = "/ p a p ers /2012.14947" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > De J a g ere t a l . , 2020 < / a >< ah re f = "/ p a p ers /2501.09208" t i tl e = "" re l = " n o f o ll o w " d a t a − t u r b o = " f a l se " c l a ss = " a ss i s t an t − l ink " x − d a t a x − t oo lt i p . r a w = "" > Kr a tt e n t ha l er , 15 J an 2025 < / a > ) :< / s t ro n g >< / l i >< / u l >< p >
5. Applications in Lattice Path Enumeration and Algebraic Combinatorics
Bicolored Motzkin paths provide a unifying combinatorial framework for enumerative and algebraic structures:
Representation-theoretic applications: The bijective correspondence with SYTs of height 5 enables enumerative interpretations in terms of representation dimension counts and branching rules (Eu et al., 2013 ).Poset theory: Interval-closed sets in products of chains are efficiently counted via bicolored Motzkin path representations, revealing polynomiality in enumeration for fixed rank (Elizalde et al., 20 Dec 2024 ).Tableaux theory: Two-rowed set-valued standard tableaux, a key object in combinatorial commutative algebra, are enumerated and structurally analyzed using bicolored Motzkin codes (Krattenthaler, 15 Jan 2025 ).Lattice walks: The direct mapping to quarter-plane and triangular walks supports the analysis of walk-in-domain models with step multiplicities and boundary constraints, with implications for weighted enumerative models (Yeats, 2014 , Courtiel et al., 2020 ).
6. Connections to Generalized and Higher Order Motzkin Paths
Bicolored Motzkin paths form a subset of ( α ⃗ , β ⃗ ) (\vec{\alpha},\vec{\beta}) ( α , β ) ( a , B ) = ( 2 , 2 ) (a,B)=(2,2) ( a , B ) = ( 2 , 2 ) DeJager et al., 2020 ):
Steps and coloring generalized to L L L k > 0 k>0 k > 0
Multivariate generating functions express colored path counts, Riordan arrays, and connections to unary–binary trees and k k k
Explicit closed-form coefficient expressions and recursive structure admit direct polynomial-combinatorial interpretations.
A plausible implication is that techniques for analyzing colored Motzkin paths generalize to richer path models, including those with amplitude constraints, step multiplicities, and nesting properties relevant in higher-dimensional combinatorial and geometric contexts.
Definition Search Book Streamline Icon: https://streamlinehq.com References (6)
