A177477
Number of permutations of 1..n avoiding adjacent step pattern up, down, up.
Original entry on oeis.org
1, 1, 2, 6, 19, 70, 331, 1863, 11637, 81110, 635550, 5495339, 51590494, 524043395, 5743546943, 67478821537, 844983073638, 11240221721390, 158365579448315, 2355375055596386, 36870671943986643, 606008531691619131, 10435226671431973345, 187860338952519968538
Offset: 0
Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and R. H. Hardin, May 10 2010 (17 terms)
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
`if`(t=3, 0, add(b(u+j-1, o-j, 2), j=1..o)))
end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25); # Alois P. Heinz, Mar 10 2020
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b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}] +
If[t == 3, 0, Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 08 2022, after Alois P. Heinz *)
A227883
Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, up.
Original entry on oeis.org
0, 0, 0, 0, 5, 50, 328, 2154, 16751, 144840, 1314149, 12735722, 134159743, 1519210786, 18272249418, 233231701166, 3159471128588, 45243728569842, 682183513506619, 10807962134238068, 179606706777512992, 3123700853586733882, 56737351453843424893
Offset: 0
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 50: 12435, 12534, 13245, ..., 52314, 52413, 53412.
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b:= proc(u, o, t) option remember;
`if`(t=7, 0, `if`(u+o=0, `if`(t in [4, 5, 6], 1, 0),
add(b(u-j, o+j-1, [1, 3, 1, 5, 6, 6][t]), j=1..u)+
add(b(u+j-1, o-j, [2, 2, 4, 4, 7, 4][t]), j=1..o)))
end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);
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b[u_, o_, t_] := b[u, o, t] =
If[t == 7, 0, If[u + o == 0, If[4 <= t <= 6, 1, 0],
Sum[b[u - j, o + j - 1, {1, 3, 1, 5, 6, 6}[[t]]], {j, 1, u}] +
Sum[b[u + j - 1, o - j, {2, 2, 4, 4, 7, 4}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)
A232899
Number of permutations of [n] cyclically avoiding the consecutive step pattern UDU (U=up, D=down).
Original entry on oeis.org
1, 1, 0, 3, 12, 35, 144, 910, 5976, 39942, 306570, 2698223, 25536132, 257563618, 2813856192, 33154390275, 415692891552, 5523237345701, 77778820305558, 1157352664763569, 18120617730892800, 297774609082108662, 5127157782095091402, 92308888110570124310
Offset: 0
a(2) = 0 because 12 and 21 do not avoid UDU (the two U's overlap).
a(3) = 3: 132, 213, 321.
a(4) = 12: 1243, 1342, 1432, 2134, 2143, 2431, 3124, 3214, 3421, 4213, 4312, 4321.
a(5) = 35: 12354, 12453, 12543, ..., 54213, 54312, 54321.
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b:= proc(u, o, t) option remember; `if`(t=4, 0,
`if`(u+o=0, `if`(t=2, 0, 1),
add(b(u+j-1, o-j, [2, 2, 4][t]), j=1..o)+
add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)))
end:
a:= n-> `if`(n<2, 1, n*b(0, n-1, 1)):
seq(a(n), n=0..30);
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b[u_, o_, t_] := b[u, o, t] = If[t == 4, 0,
If[u + o == 0, If[t == 2, 0, 1],
Sum[b[u + j - 1, o - j, {2, 2, 4}[[t]]], {j, 1, o}] +
Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}]]];
a[n_] := If[n < 2, 1, n b[0, n - 1, 1]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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