Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two polynomials $f(q)$ and $g(q)$, we write $f(q)\geq g(q)$ provided that $f(q)-g(q)$ is a polynomial having non-negative coefficients.

I would like to ask:

QUESTION. Suppose $n<m, j<k, 2j<m$ and $0\leq i\leq j$. Is it true that whenever $\binom{n}k_q\geq\binom{m}j_q$ then $\binom{n}{k-i}_q\geq\binom{m}{j-i}_q$?

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two polynomials $f(q)$ and $g(q)$, we write $f(q)\geq g(q)$ provided that $f(q)-g(q)$ is a polynomial having non-negative coefficients.

I would like to ask:

QUESTION. Suppose $n<m, j<k, 2j<m$ and $2k<n$. Is it true that whenever $\binom{n}k_q\geq\binom{m}j_q$ then $\binom{n}{k-1}_q\geq\binom{m}{j-1}_q$?

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two polynomials $f(q)$ and $g(q)$, we write $f(q)\geq g(q)$ provided that $f(q)-g(q)$ is a polynomial having non-negative coefficients.

I would like to ask:

QUESTION. Suppose $2j<m, j<k$ and $0\leq i\leq j$. If $\binom{n}k_q\geq\binom{m}j_q$ then is it true that $\binom{n}{k-i}_q\geq\binom{m}{j-i}_q$?

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two polynomials $f(q)$ and $g(q)$, we write $f(q)\geq g(q)$ provided that $f(q)-g(q)$ is a polynomial having non-negative coefficients.

I would like to ask:

QUESTION. Suppose $n<m, j<k, 2j<m$ and $0\leq i\leq j$. Is it true that whenever $\binom{n}k_q\geq\binom{m}j_q$ then $\binom{n}{k-i}_q\geq\binom{m}{j-i}_q$?