Sum with products turned into subsequences

Question

Let $p, q \in \mathbb{Z}$.

Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and

$$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$$

Then we have an integer sequence given by \begin{align} a(0, m)& = 1\\ a(n, m)& = \sum\limits_{j=0}^{2^{\operatorname{wt}(n)}-1}m^{\operatorname{wt}(n)-\operatorname{wt}(j)}\prod\limits_{k=0}^{\operatorname{wt}(n)-1}(1+\operatorname{wt}(\left\lfloor\frac{j}{2^k}\right\rfloor))^{t_{k+1}+1} \end{align} I conjecture that $$a(n, -1) = \sum\limits_{j=0}^{2^{\operatorname{wt}(n)}-1}(-1)^{\operatorname{wt}(n)-\operatorname{wt}(j)}a(f(n,j),0)$$ and $$a(n, 0) = \sum\limits_{j=0}^{2^{\operatorname{wt}(n)}-1}a(f(n,j),-1)$$ where $a(n,-1)$ is A329369, number of permutations of ${1,2,...,m}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)...b(1)b(0)$ ($0 \leqslant k < m-1$) is the binary expansion of $n$. The excedance set of a permutation $p$ of ${1,2,...,m}$ is the set of indices $i$ such that $p(i) > i$; it is a subset of ${1,2,...,m-1}$.

and $a(n,0)$ is A284005, \begin{align} a(0, 0)& = 1\\ a(n,0)& = (1+\operatorname{wt}(n))a(\left\lfloor\frac{n}{2}\right\rfloor,0) \end{align}

and finally $f(n,k)$ is A295989, irregular triangle $T(n, k)$, read by rows, $n \geqslant 0$ and $0 \leqslant k <$ A001316$(n)$: $T(n, k)$ is the $(k+1)$-th nonnegative number $m$ such that $n \operatorname{AND} m = m$ (where $\operatorname{AND}$ denotes the bitwise $\operatorname{AND}$ operator).

\begin{align} T(n, 0)& = 0\\ T(2n, k)& = 2T(n,k)\\ T(2n+1, 2k)& = 2T(n,k)\\ T(2n+1, 2k+1)& = 2T(n,k) + 1 \end{align}

In other words

$$a(n, -1) = \sum\limits_{j=0}^{n}(-1)^{\operatorname{wt}(n)-\operatorname{wt}(j)}(\binom{n}{j}\operatorname{mod} 2)a(j,0)$$ and $$a(n, 0) = \sum\limits_{j=0}^{n}(\binom{n}{j}\operatorname{mod} 2)a(j,-1)$$

Is there a way to prove it?

Answer 1

Actually, these two conjectures are actually equivalent.

We need a lemma: Lucas' Theorem: $\binom{a}{b}$ is odd if and only if $a\&b=b$ where $\&$ is bitwise AND operation. Let $S_j$ be the set of integers $i$ such that $i\&j=i$. Thus, if the first relation

$$a(n, -1) = \sum\limits_{j=0}^{n}(-1)^{\operatorname{wt}(n)-\operatorname{wt}(j)}(\binom{n}{j}\operatorname{mod} 2)a(j,0)$$ holds, we have $$a(n, -1) = \sum\limits_{j\in S_n}(-1)^{\operatorname{wt}(n)-\operatorname{wt}(j)}a(j,0)$$ And therefore, $$\sum_{j=0}^{n}(\binom{n}{j}\operatorname{mod} 2)a(j,-1)=\sum_{j\in S_n}a(j,-1)=\sum_{j\in S_n}\sum_{k\in S_j}(-1)^{\text{wt}(j)-\text{wt}(k)}a(k,-1)$$ Swapping the sums, we have the right hand side equals to $$\sum_{k\in S_n}a(k,-1)\sum_{j\text{ such that }k\in S_j,j\in S_n}(-1)^{\text{wt}(j)-\text{wt}(k)}$$ All the other $k\in S_n$ vanishs the sum $\sum_{j\text{ such that }k\in S_j,j\in S_n}(-1)^{\text{wt}(j)-\text{wt}(k)}$ because for every bit that is $1$ in $n$ and $0$ in $k$, it contribute a factor $1$ if that bit in $j$ is $0$ and $-1$ if that bit in $j$ is $1$. Therefore, the sum $$\sum_{k\in S_n}a(k,-1)\sum_{j\text{ such that }k\in S_j,j\in S_n}(-1)^{\text{wt}(j)-\text{wt}(k)}$$ is actually $a(n,-1)$, which implies the conjecture $a(n, 0) = \sum\limits_{j=0}^{n}(\binom{n}{j}\operatorname{mod} 2)a(j,-1)$. Similarly, we can derive the first conjecture using the second.

So, we only need to proof $a(n, 0) = \sum\limits_{j=0}^{n}(\binom{n}{j}\operatorname{mod} 2)a(j,-1)=\sum_{j\in S_n}a(j,-1)$.

Notice that the sum $\sum_{j\in S_n}a(j,-1)$, if we take the combinatorial argument as granted, it counts the following permutations: for all the $k${th} bit of $n$ that is $0$, $\sigma(k)\le k$. This is because all the other bits when summed, cancelled out the restrictions. (If the $k$th bit of $n$ is $1$, that all the $S_k$ with $k$th bit $1$ with have $\sigma(k)> k$, and all the $S_k$ with $k$th bit $0$ with have $\sigma(k)\le k$. So These two cancelled out if we sum the elements in $S_n$.) So we claim the number of such permutations is $a(n,0)$.

Actually, we can count the permutations as follows: for the $0$ bits of $n$ (call it $i_1,i_2,\dots,i_k$), we select the permutation one by one. We select $\sigma(i_l)$ to be one of $1,2,\dots,i_l$ except $\sigma(i_1),\dots,\sigma(i_{l-1})$. So this is $i_l-(\text{number of zeroes before $i_l$})$, that is, the number of ones before $i_l$. The rest of them we choose it freely, so it is $(\text{number of ones})!$. That is, we can assign $t$ to the $t$th one in the binary representation of $n$, or, the number of $1$ not after that bit, and take the product. Let $n=b_rb_{r-1}\dots b_0$ be its binary representation. So, the overall number of permutation is $\prod_{u=1}^r (\text{the number of ones not later than digit }b_u)$, and thus equals $a(n,0)$ (because the combinatorial meaning of $a(n,0)$ is such.)