The <i>q</i>-log-concavity of <i>q</i>-ballot numbers

doi:10.3868/s140-DDD-024-0015-x

Front. Math. China

2024, Vol. 19

Issue (5) : 247-254 https://doi.org/10.3868/s140-DDD-024-0015-x

The q-log-concavity of q-ballot numbers

Xinmiao LIU, Jiangxia HOU(

), Fengxia LIU

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

Download: PDF(827 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Carlitz and Riordan introduced the q-analogue $f q (n, k)$ of ballot numbers. In this paper, using the combinatorial interpretation of $f q (n, k)$ and constructing injections, we prove that $f q (n, k)$ is q-log-concave with respect to $n$ and $k$ , i.e., all coefficients of the polynomials $f q (n, k) 2 − f q (n + 1, k) f q (n − 1, k)$ and $f q (n, k) 2 − f q (n, k + 1) f q (n, k − 1)$ are non-negative for $0 < k < n$ .

Keywords q-log-concavity q-ballot number lattice path inversion

Corresponding Author(s): Jiangxia HOU

Online First Date: 15 October 2024 Issue Date: 31 October 2024

Cite this article:

Xinmiao LIU,Jiangxia HOU,Fengxia LIU. The q-log-concavity of q-ballot numbers[J]. Front. Math. China, 2024, 19(5): 247-254.

URL:

https://academic.hep.com.cn/fmc/EN/10.3868/s140-DDD-024-0015-x
https://academic.hep.com.cn/fmc/EN/Y2024/V19/I5/247

Fig.1 Lattice path contained in

P 2, 1

n	k
n	0	1	2	3
0	1
1	1	$q$
2	1	$q + q 2$	$q 2 + q 3$
3	1	$q + q 2 + q 3$	$q 2 + q 3 + 2 q 4 + q 5$	$q 3 + q 4 + 2 q 5 + q 6$

Tab.1

f q (n, k), 0 ≤ k ≤ n ≤ 3

Fig.2 Injection from (11211221, 121212) to (1121122, 1212121)

Fig.3 Injection from (11121122, 121212) to (1112212, 1211122)

	1	2
$π R$	$a + 1$	$i ? a$
$σ R$	$a$	$i ? a$
$π L$	$n ? a$	$k ? i + a$
$σ L$	$n ? a + 1$	$k ? i + a$

Tab.2 Count of digits 1 and 2 in

π L, π R, σ L

, and

σ R

Fig.4 Injection from (112211212, 1121112) to (11221121, 11211122)

Fig.5 Injection from (112212121, 1121112) to (11221112, 11212121)

1	L M Butler. The q-log-concavity of q-binomial coefficients. J Combin Theory Ser A 1990; 54(1): 54–63
2	L Carlitz. Sequences, paths, ballot numbers. Fibonacci Quart 1972; 10(5): 531–549
3	L Carlitz, J Riordan. Two element lattice permutation numbers and their q-generalization. Duke Math J 1964; 31(3): 371–388
4	F Chapoton, J Zeng. A curious polynomial interpolation of Carlitz–Riordan’s q-ballot numbers. Contrib Discrete Math 2015; 10(1): 99–112
5	W Y C Chen, L X W Wang, A L B Yang. Schur positivity and the q-log-convexity of the Narayana polynomials. J Algebraic Combin 2010; 32(3): 303–338
6	L Comtet. Advanced Combinatorics. Dordrecht: D Reidel, 1974
7	K Q Ji. The q-log-concavity and unimodality of q-Kaplansky numbers. Discrete Math 2022; 345(6): 112821
8	B E Sagan. Log concave sequences of symmetric functions and analogs of the Jacobi–Trudi determinants. Trans Amer Math Soc 1992; 329(2): 795–811
9	B E Sagan. Inductive proofs of q-log concavity. Discrete Math 1992; 99(1/2/3): 298–306
10	S Spiro. Ballot permutations and odd order permutations. Discrete Math 2020; 343(6): 111869
11	R P Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In: Graph Theory and Its Applications: East and West (Jinan, 1986), Annals of the New York Academy of Sciences, Vol 576. New York: New York Acad Sci, 1989, 500–535
12	D G L Wang, J J R Zhang. A Toeplitz property of ballot permutations and odd order permutations. Electron J Combin 2022; 27(2): 2.55
13	D G L Wang, T Y Zhao. The peak and descent statistics over ballot permutations. Discrete Math 2022; 345(3): 112739

[1]	Jishe FENG, Xiaomeng WANG, Xiaolu GAO, Zhuo PAN. The research and progress of the enumeration of lattice paths[J]. Front. Math. China, 2022, 17(5): 747-766.

Viewed

Full text

Abstract

Cited

Shared

Discussed