Abstract
We explore the combinatorial properties of the set
S
(
k
,
ℓ
;
n
)
$\mathfrak {S}(k,\ell; n)$
German upper S left parenthesis k comma script l semicolon n right parenthesis
consisting of standard Young tableaux with specific shape constraints. Using bijections with Motzkin and free Motzkin paths, we enumerate
S
m
(
3
,
0
;
n
)
$\mathrm {S}_m(3,0;n)$
normal upper S Subscript m Baseline left parenthesis 3 comma 0 semicolon n right parenthesis
and
S
m
(
2
,
1
;
n
)
$\mathrm {S}_m(2,1;n)$
normal upper S Subscript m Baseline left parenthesis 2 comma 1 semicolon n right parenthesis
for all
m
, and
S
m
(
0
,
3
;
n
)
$\mathrm {S}_m(0,3;n)$
normal upper S Subscript m Baseline left parenthesis 0 comma 3 semicolon n right parenthesis
and
S
m
(
1
,
2
;
n
)
$\mathrm {S}_m(1,2;n)$
normal upper S Subscript m Baseline left parenthesis 1 comma 2 semicolon n right parenthesis
for small values of
m
. We derive explicit formulae such as
S
2
(
3
,
0
;
n
)
=
M
n
−
M
n
−
1
$\mathrm {S}_2(3,0;n)=M_n - M_{n-1}$
normal upper S 2 left parenthesis 3 comma 0 semicolon n right parenthesis equals upper M Subscript n Baseline minus upper M Subscript n minus 1
and
S
4
(
3
,
0
;
n
)
=
M
n
−
3
$\mathrm {S}_4(3,0;n)=M_{n-3}$
normal upper S 4 left parenthesis 3 comma 0 semicolon n right parenthesis equals upper M Subscript n minus 3
, where
M
n
$M_n$
upper M Subscript n
denotes the number of Motzkin paths of length
n
. Furthermore, we offer an alternative proof of Regev’s identity
S
(
2
,
1
;
n
)
=
1
+
S
odd
(
2
,
1
;
n
+
1
)
$\mathrm {S}(2,1;n)=1+\mathrm { S}_{\mathrm {odd}}(2,1;n+1)$
normal upper S left parenthesis 2 comma 1 semicolon n right parenthesis equals 1 plus normal upper S Subscript odd Baseline left parenthesis 2 comma 1 semicolon n plus 1 right parenthesis
, thereby deepening the connection between standard Young tableaux and lattice path combinatorics.