OFFSET
0,3
COMMENTS
a(n) = number of ordered pairs of permutations of [n] such that the first has an ascent wherever the second has a descent and vice versa. For example, the pair of permutations (1243, 4123) does not qualify because they have a common ascent starting at location 2, and a(2) = 2 counts (12, 21), (21, 12). - David Callan, Sep 15 2013
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..250 (terms 0..150 from Alois P. Heinz)
FORMULA
a(n) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^2. - Alois P. Heinz, Sep 15 2020
a(n) ~ c * d^n * n!^2, where d = 0.552406011965766199179395470003589240257321... and c = 1.6412834540969426814342654061364... - Vaclav Kotesovec, Sep 18 2020
EXAMPLE
a(1)=1^2; a(2)=1^2+1^2; a(3)=1^2+2^2+2^2+1^2; a(4)=1^2+3^2+5^2+3^2+3^2+5^2+3^2+1^2.
MAPLE
ct := proc(k) option remember; local i, out, n; if k=0 then RETURN(1); fi; n := floor(evalf(log[2](k)))+1; if k=2^n or k=2^(n+1)-1 then RETURN(1); fi; out := 0; for i from 1 to n do if irem(iquo(k, 2^(i-1)), 2) = 1 and irem(iquo(2*k, 2^(i-1)), 2) =0 then out := out+(n-1)!/(i-1)!/(n-i)!* ct(floor(irem(k, 2^(i-1))+2^(i-2)))*ct(iquo(k, 2^i)); fi; od; out; end: seq(add(ct(i)^2, i=floor(2^(n-1))..2^n-1), n=0..15);
# second Maple program:
b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
end:
a:= n-> b(0, n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 02 2015
MATHEMATICA
b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1, Sum[Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}], {j, 1, u}] + Sum[Sum[b[u + j - 1, o - j, h - i], {i, 1, h}], {j, 1, o}]]; a[n_] := b[0, n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mike Zabrocki, Mar 31 2001
EXTENSIONS
Two more terms from Max Alekseyev, May 06 2009
a(0) prepended, a(18) from Alois P. Heinz, Jul 02 2015
STATUS
approved