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A158432
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Number of permutations of 1..n containing the relative rank sequence { 45312 } at any spacing.
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3
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1, 26, 458, 6996, 101072, 1438112, 20598112, 300892896, 4521034917, 70286670034, 1135485759114, 19121776482564, 336412530327804, 6191800556586104, 119301546930406184, 2406376964044265344, 50786085223779295344, 1120447461653440780128, 25810064637612342838624
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OFFSET
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5,2
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COMMENTS
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Same series for 54321 12345 45321 21345 12354 54312 34521 32145 12543 54123 23451 43215 15432 51234 21354 34512 32154 21543 45123.
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LINKS
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FORMULA
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MAPLE
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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:= (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))):
a:= n-> n! -g(n, 4, []):
# second Maple program
a:= proc(n) option remember; `if`(n<5, 0, `if`(n=5, 1,
((132-142*n-301*n^2-35*n^3+25*n^4+n^5)*a(n-1)
-2*(10*n^3+33*n^2-181*n-2)*(n-1)^2*a(n-2)
+64*(n-2)^2*(n-1)^3*a(n-3))/ ((n+4)*(n-5)*(n+3)^2)))
end:
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MATHEMATICA
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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, 4, {}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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