$0<a_1<a_2<\cdots<a_k$ satisfy $\bigoplus\limits_{i=1}^ka_i = 2^n-1$ ($\oplus$ represents bitwise or) prove |config number of k is even - odd| =1

Question

In a finite set $S$ with $|S|=n$, let $N_n^k$ be the number of ascending chains of $k$ nonvoid subsets $\emptyset \ne S_1 \subsetneq S_2 \subsetneq \cdots \subsetneq S_k$ with strict inclusion such that $\bigcup_k S_k=S$;

show that $|\sum_{\text{odd} \ k} N_n^k - \sum_{\text{even} \ k} N_n^k|=1$

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I want to prove this (informally) prove that number of acyclic orientations of a graph is the chromatic polynomial evaluated at -1. in another way, but I encountered a difficulty.

I don't know if it's correct and I cannot prove it:

$$ \sum _{\varnothing \neq J\subseteq(S \cap Q)}(-1)^{|J|+1}|g(S,J)| = \sum _{\varnothing \neq J\subseteq \{x|x\subseteq(S \cap Q),x\not=\varnothing\}}(-1)^{|J|+1}\left|g(S,\bigcup_{j\in J}j)\right| $$

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$0<a_1<a_2<\cdots<a_k$ satisfy $\bigoplus\limits_{i=1}^ka_i = 2^n-1$ ($\oplus$ represents bitwise or)

There are many arrays as and ks that satisfy this condition,

And I find $|\texttt{config number when k is even} - \texttt{config number when k is odd}|$ = 1 is correct when $n\leq5$.

Is that correct for all positive integer n?

For example, when $n=2$,

$(11)_2 = (11)_2$

k=1,1 config

$(01)_2 \oplus (11)_2 = (11)_2$

$(10)_2 \oplus (11)_2 = (11)_2$

$(01)_2 \oplus (10)_2 = (11)_2$

k=2,3 configs

$(01)_2 \oplus (10)_2 \oplus (11)_2 = (11)_2$

k=3,1 config

So $|1-3+1| = 1$.

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I've also written a messy code to verify it when $n\leq5$: (n=4 in the code)

#include <bits/stdc++.h>
using namespace std;
int m=16,c[50];
void dfs(int u,int orsum,int count) {
    if(u==m) { if(orsum==m-1) c[count]++; return ; }
    dfs(u+1,orsum|u,count+1);
    dfs(u+1,orsum,count);
}
int main(){
    dfs(1,0,0);
    int x=-1,i;
    int as=0;
    for(i=0;i<=m;i++) printf("%d element(s) %d arg %d\n",i,c[i],x),as+=c[i]*x,x*=-1;
    printf("%d",as); 
    return 0;
}

Output (If this helps)

0 element(s) 0 arg -1
1 element(s) 1 arg 1
2 element(s) 39 arg -1
3 element(s) 321 arg 1
4 element(s) 1225 arg -1
5 element(s) 2919 arg 1
6 element(s) 4977 arg -1
7 element(s) 6431 arg 1
8 element(s) 6435 arg -1
9 element(s) 5005 arg 1
10 element(s) 3003 arg -1
11 element(s) 1365 arg 1
12 element(s) 455 arg -1
13 element(s) 105 arg 1
14 element(s) 15 arg -1
15 element(s) 1 arg 1
16 element(s) 0 arg -1