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A159139
Number of permutations of 1..n containing the relative rank sequence { 213465 } at any spacing.
3
1, 37, 891, 18043, 337210, 6081686, 108469917, 1941309261, 35187952132, 649951312000, 12286366975723, 238445927000811, 4762398793018878, 98074791689121162, 2085684931155975120, 45859509146309390064, 1043533983233372354613, 24590543663448304800169
OFFSET
6,2
COMMENTS
Same series for 654321 123456 564321 213456 123465 654312 456321 321456 123654 654123 345621 432156 126543 651234 564312 456312 321465 213654 564123 345612 432165 216543 561234 234561 543216 165432 612345 456123 321654.
LINKS
Eric Weisstein's World of Mathematics, Permutation Pattern
FORMULA
a(n) = A214152(n,6) = A000142(n)-A047890(n) = A000142(n)-A214015(n,5). - Alois P. Heinz, Jul 05 2012
MAPLE
h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
+add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l)
`if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
end:
a:= n-> n! -g(n, 5, []):
seq(a(n), n=6..30); # Alois P. Heinz, Jul 05 2012
# Alternative:
a:= proc(n) option remember; `if`(n<6, 0, `if`(n=6, 1,
((2475-4819*n^2-2985*n+175*n^4-1021*n^3+n^6+49*n^5)*a(n-1)
-(35*n^4+441*n^3-845*n^2-4147*n-489)*(n-1)^2*a(n-2)
+(-1668+329*n+259*n^2)*(n-1)^2*(n-2)^2*a(n-3)
-225*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4))/ ((n-6)*(n+6)^2*(n+4)^2)))
end:
seq(a(n), n=6..30); # Alois P. Heinz, Sep 26 2012
MATHEMATICA
h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1 &, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
a[n_] := n! - g[n, 5, {}];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Jun 19 2018, from first Maple program *)
KEYWORD
nonn
AUTHOR
R. H. Hardin, Apr 05 2009
EXTENSIONS
More terms from Alois P. Heinz, Jul 05 2012
STATUS
approved