A346204 a(n) is the number of permutations on [n] with at least one strong fixed point and at least one small descent.

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

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 10 2021

Keywords

• nonn
• more

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			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.
		

References

• E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

• M. Lind, E. Fiorini, A. Woldar, and W. H. T. Wong, On Properties of Pebble Assignment Graphs, Journal of Integer Sequences, 24(6), 2020.

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346198, A346199.

Programs

Python

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)

Formula

a(n) = A006932(n) - A346199(n).