Search: a117106 -id:a117106
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A279571
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Number of length n inversion sequences avoiding the patterns 100, 101, and 201.
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+10
23
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1, 1, 2, 6, 22, 92, 424, 2106, 11102, 61436, 353980, 2110366, 12955020, 81569168, 525106698, 3447244188, 23028080268, 156246994264, 1075127143948, 7492458675666, 52820934349420, 376331681648402, 2707312468516446, 19650530699752470, 143807774782994412, 1060472244838174574, 7875713244761349666, 58876660310205135380, 442862775457168812898, 3350397169412102710198
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OFFSET
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0,3
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COMMENTS
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A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i<j<k) such that e_i > e_j <= e_k and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 100, 101, and 201.
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LINKS
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EXAMPLE
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The length 4 inversion sequences avoiding (100,101,201) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0102, 0103, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
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MAPLE
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b:= proc(n, i, s, m) option remember;
`if`(n=0, 1, add(b(n-1, i+1, s minus {$j..m-
`if`(j=m, 1, 0)} union {i+1}, max(m, j)), j=s))
end:
a:= n-> b(n, 1, {1}, 0):
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MATHEMATICA
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b[n_, i_, s_, m_] := b[n, i, s, m] = If[n == 0, 1, Sum[b[n-1, i+1, s ~Complement~ Range[j, m - If[j == m, 1, 0]] ~Union~ {i+1}, Max[m, j]], {j, s}]];
a[n_] := b[n, 1, {1}, 0];
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CROSSREFS
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Cf. A000108, A057552, A108307, A117106, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279572, A279573.
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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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A363682
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Number of plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) with n edges.
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+10
0
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1, 1, 1, 2, 5, 12, 32, 93, 279, 872, 2830, 9433, 32223, 112527, 400370, 1448520, 5320023, 19802827, 74612164, 284238390, 1093757436, 4247742956, 16636921148, 65671960544, 261111950308, 1045172796381, 4209807155949, 17055625810984, 69476952146529, 284467866640048
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OFFSET
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1,4
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COMMENTS
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a(n) is also the number of walks of length n-1 in the quadrant, starting and ending at the origin, with step-set {0,E,S,NW,SE} (where 0 is the stay-step).
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LINKS
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MAPLE
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A:=proc(n, i, j) option remember:
if n=0 and i=0 and j=0 then return 1:
elif n<=0 or j<0 or i<0 then return 0:
else
return A(n-1, i, j)+A(n-1, i-1, j)+A(n-1, i, j+1)+A(n-1, i+1, j-1)+A(n-1, i-1, j+1):
fi:
end proc:
seq(A(n-1, 0, 0), n=1..20);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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