A291685
Number of permutations p of [n] such that 0p has a nonincreasing jump sequence.
Original entry on oeis.org
1, 1, 2, 5, 16, 52, 189, 683, 2621, 10061, 40031, 159201, 650880, 2657089, 11062682, 46065143, 194595138, 822215099, 3513875245, 15021070567, 64785349064, 279575206629, 1214958544538, 5283266426743, 23106210465665, 101120747493793, 444614706427665
Offset: 0
a(3) = 5 = 6 - 1 counts all permutations of {1,2,3} except 132 with jump sequence 1, 2, 1.
Row sums and main diagonal (shifted) of
A291684.
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, j), j=1..min(t, u))+
add(b(u+j-1, o-j, j), j=1..min(t, o)))
end:
a:= n-> b(0, n$2):
seq(a(n), n=0..30);
-
b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
Sum[b[u-j, o+j-1, j], {j, Min[t, u]}]+
Sum[b[u+j-1, o-j, j], {j, Min[t, o]}]];
a[n_] := b[0, n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)
A288910
Number of permutations p of [n] such that 0p has a nonincreasing up-jump sequence and also has a nonincreasing down-jump sequence.
Original entry on oeis.org
1, 1, 2, 5, 18, 69, 303, 1357, 6552, 31961, 163587, 839710, 4485686, 23917300, 131366017, 722130351, 4059017880, 22809880116, 130713878478, 748596353814, 4354695496124, 25349532110660, 149415724294027, 881419904003486, 5256588077063477, 31377362011756061
Offset: 0
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b:= proc(u, o, t, s) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, j, s), j=1..min(t, u))+
add(b(u+j-1, o-j, t, j), j=1..min(s, o)))
end:
a:= n-> b(0, n$3):
seq(a(n), n=0..26);
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b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, 1,
Sum[b[u - j, o + j - 1, j, s], {j, Min[t, u]}] +
Sum[b[u + j - 1, o - j, t, j], {j, Min[s, o]}]];
a[n_] := b[0, n, n, n];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)
A288912
Number of permutations p of [n] such that 0p has a nonincreasing down-jump sequence.
Original entry on oeis.org
1, 1, 2, 6, 23, 106, 558, 3284, 21200, 148539, 1119273, 9013112, 77106652, 697811164, 6652604804, 66593158893, 697756930786, 7633155361594, 86969814549075, 1029939820075074, 12652809025029242, 160977119684852369, 2117642963178349336, 28763717105362639324
Offset: 0
-
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, j), j=1..min(t, u))+
add(b(u+j-1, o-j, t), j=1..o))
end:
a:= n-> b(0, n$2):
seq(a(n), n=0..30);
-
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
Sum[b[u - j, o + j - 1, j], {j, Min[t, u]}] +
Sum[b[u + j - 1, o - j, t], {j, o}]];
a[n_] := b[0, n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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