#Python program for A368054 by John Rascoe 1/15/2024
from itertools import permutations, combinations
from collections import Counter

def A3680054(row_n):
    C,b = Counter(),[]
    p = list(permutations(list(range(1,(2*row_n)+1))))
    #Sieves out perms that have alternating parity starting odd and puts them in list b
    for k in range(len(p)):
        c = 0
        for j in range(len(p[k])):
            if p[k][j]%2 == (j+1)%2:
                c += 1
            else:
                break
        if c == (2*row_n):
            b.append(p[k])
    for k in range(len(b)):
        cx,X = 0,[]
        #internal loop for a given partition
        #first puts all pairs of terms with absolute difference > 1 in list X
        for j in range(1,len(b[k])):
            x,y = 0,0
            if abs(b[k][j] - b[k][j-1]) > 1:
                x += b[k][j]
                y += b[k][j-1]
                X.append((y,x))
                #first end point
                if y%2 != b[k][0]%2:
                    if x > y and y < b[k][0] < x: 
                        cx += 1
                    elif x < b[k][0] < y:
                        cx += 1
                #last endpoint
                if x%2 != b[k][-1]%2:
                    if x < y and x < b[k][-1] < y:
                        cx += 1
                    elif y < b[k][-1] < x:
                        cx += 1
        #finds internal crossings
        if len(X) > 0:
            Xp = list(combinations(X,2))
            for k1 in range(len(Xp)):
                a = Xp[k1][0][0]
                c = Xp[k1][1][0]
                e = min(Xp[k1][0])
                f = max(Xp[k1][0])
                g = min(Xp[k1][1])
                h = max(Xp[k1][1])
                if (a%2) == (c%2):
                    if (g < e < h and h < f) or (g < f < h and e < g):
                        cx += 1
        C.update({cx})
        #optional printout below for all partitions in a row and how many crossings for each
        #print("Partition",b[k],"has",cx,"crossings")
    zz = list(C[i] for i in range(0,max(C)+1))
    return(zz)
                
for i in range(5):
    print(A3680054(i))