A362603 Number of permutations p of [2n] in which exactly the first n terms satisfy the up-down property p(1) < p(2) > p(3) < ... .

1, 1, 4, 90, 3024, 176400, 14731200, 1710268560, 261131270400, 50881298307840, 12308045700787200, 3620112665116147200, 1272148028456410828800, 526419950201914728960000, 253357552054376603817984000, 140324455080520735061157120000, 88618646911930055808757309440000

Offset: 0

Author

Alois P. Heinz, Apr 27 2023

Keywords

• nonn

Examples

			a(0) = 1: (), the empty permutation.
a(1) = 1: 21.
a(2) = 4: 1234, 1243, 1342, 2341.
a(3) = 90: 143256, 143265, 153246, 153264, ..., 564213, 564231, 564312, 564321.
a(4) = 3024: 13245678, 13245687, 13245768, 13245786, ..., 78456213, 78456231, 78456312, 78456321.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..234

Crossrefs

Cf. A000111, A001813, A092580.

Programs

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> (2*n)!/n!*`if`(n=0, 1, b(n, 0)-b(n+1, 0)/(n+1)):
seq(a(n), n=0..19);

Python

from fractions import Fraction
from math import factorial
from itertools import count, islice, accumulate
def A362603_gen(): # generator of terms
    yield 1
    blist, c = (0,1), 1
    for n in count(1):
        blist, a, c = tuple(accumulate(reversed(blist),initial=0)), blist[-1], c*((n<<2)-2)
        yield int(c*(a-Fraction(blist[-1],(n+1))))
A362603_list = list(islice(A362603_gen(),20)) # Chai Wah Wu, Apr 28 2023

Formula

a(n) = (2*n)!/n! * (A000111(n) - A000111(n+1)/(n+1)) for n > 0, a(0) = 1.

a(n) = A092580(2n,n).