Xufei Liu has authored 4 sequences.
A346199
a(n) is the number of permutations on [n] with at least one strong fixed point and no small descents.
Original entry on oeis.org
1, 1, 1, 5, 19, 95, 569, 3957, 31455, 281435, 2799981, 30666153, 366646995, 4751669391, 66348304849, 992975080813, 15856445382119, 269096399032035, 4836375742967861, 91766664243841393, 1833100630242606203, 38452789552631651191, 845116020421125048153
Offset: 1
For n = 4, the a(4) = 5 permutations on [4] with strong fixed points but no small descents: {(1*, 2*, 3*, 4*), (1*, 3, 4, 2), (1*, 4, 2, 3), (2, 3, 1, 4*), (3, 1, 2, 4*)} where * marks strong fixed points.
- E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.
A346198
a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.
Original entry on oeis.org
0, 1, 1, 8, 43, 283, 2126, 17947, 168461, 1741824, 19684171, 241506539, 3198239994, 45482655683, 691471698917, 11193266251700, 192238116358427, 3491633681792507, 66875708261486766, 1347168876070616179, 28474546456352896021, 630130731702950549248, 14570725407559756078387, 351411668456841530417027
Offset: 1
For n = 4, the a(4) = 8 permutations on [4] with no strong fixed points but has small descents: {([2, 1], [4, 3]), (2, [4, 3], 1), ([3, 2], 4, 1), (3, 4, [2, 1]), (4, 1, [3, 2]), (4, [2, 1], 3), ([4, 3], 1, 2), (<4, 3, 2, 1>)} []small descent, <>consecutive small descents.
- E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.
A346189
a(n) is the number of permutations on [n] with no strong fixed points or small descents.
Original entry on oeis.org
0, 0, 2, 6, 34, 214, 1550, 12730, 116874, 1187022, 13219550, 160233258, 2100360778, 29610224590, 446789311934, 7185155686666, 122690711149290, 2217055354281582, 42269657477711198, 847998698508705834, 17857221256001240458, 393839277313540073230, 9078806210245773668990, 218340709713567352161226
Offset: 1
For n = 4, the a(4) = 6 permutations on [4] with no strong fixed points or small descents: {(2,3,4,1),(3,4,1,2),(4,1,2,3),(3,1,4,2),(2,4,1,3),(4,2,3,1)}.
- E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.
A346204
a(n) is the number of permutations on [n] with at least one strong fixed point and at least one small descent.
Original entry on oeis.org
0, 0, 2, 5, 24, 128, 795, 5686, 46090, 418519, 4213098, 46595650, 561773033, 7333741536, 103065052300, 1551392868821, 24902155206164, 424588270621876, 7663358926666175, 145967769353476594, 2926073829112697318, 61577929208485406331, 1357369100658321844470, 31276096500003460511422
Offset: 1
For n=4, the a(4)=5 permutations on [4] with strong fixed points and small descents: {(1*, 2*, [4, 3]), (1*, [3, 2], 4*), (1*, <4, 3, 2>), ([2, 1], 3*, 4*), (<3, 2, 1>, 4*)}. *strong fixed point, []small descent, <>consecutive small descents.
- E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.
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import math
bn = [1,1,1]
wn = [0,0,0]
kn = [1,1,1]
def summation(n):
final = bn[n] - bn[n-1]
for k in range(4,n+1):
final -= wn[k-1]*bn[n-k]
return final
def smallsum(n):
final = bn[n-1]
for k in range(4,n+1):
final += wn[k-1]*bn[n-k]
return final
def derrangement(n):
finalsum = 0
for i in range(n+1):
if i%2 == 0:
finalsum += math.factorial(n)*1//math.factorial(i)
else:
finalsum -= math.factorial(n)*1//math.factorial(i)
if finalsum != 0:
return finalsum
else:
return 1
def fixedpoint(n):
finalsum = math.factorial(n-1)
for i in range(2,n):
finalsum += math.factorial(i-i)*math.factorial(n-i-1)
print(math.factorial(i-i)*math.factorial(n-i-1))
return finalsum
def no_cycles(n):
goal = n
cycles = [0, 1]
current = 2
while current<= goal:
new = 0
k = 1
while k<=current:
new += (math.factorial(k-1)-cycles[k-1])*(math.factorial(current-k))
k+=1
cycles.append(new)
current+=1
return cycles
def total_func(n):
for i in range(3,n+1):
bn.append(derrangement(i+1)//(i))
kn.append(smallsum(i))
wn.append(summation(i))
an = no_cycles(n)
tl = [int(an[i]-kn[i]) for i in range(n+1)]
factorial = [math.factorial(x) for x in range(0,n+1)]
print("A346189 :" + str(wn[1:]))
print("A346198 :" + str([factorial[i]-wn[i]-tl[i]-kn[i] for i in range(n+1)][1:]))
print("A346199 :" + str(kn[1:]))
print("A346204 :" + str(tl[1:]))
total_func(20)
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