A291684
Number T(n,k) of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 5, 5, 0, 1, 9, 12, 14, 16, 0, 1, 17, 36, 36, 47, 52, 0, 1, 31, 81, 98, 117, 166, 189, 0, 1, 57, 174, 327, 327, 425, 627, 683, 0, 1, 101, 413, 788, 988, 1116, 1633, 2400, 2621, 0, 1, 185, 889, 1890, 3392, 3392, 4291, 6471, 9459, 10061
Offset: 0
T(3,1) = 1: 123.
T(3,2) = 2: 213, 231.
T(3,3) = 2: 312, 321.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 2;
0, 1, 5, 5, 5;
0, 1, 9, 12, 14, 16;
0, 1, 17, 36, 36, 47, 52;
0, 1, 31, 81, 98, 117, 166, 189;
0, 1, 57, 174, 327, 327, 425, 627, 683;
0, 1, 101, 413, 788, 988, 1116, 1633, 2400, 2621;
Columns k=0-10 give:
A000007,
A057427,
A292168,
A292169,
A292170,
A292171,
A292172,
A292173,
A292174,
A292175,
A292176.
Row sums and T(n+1,n+1) give
A291685.
-
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:
T:= (n, k)-> b(0, n, k)-`if`(k=0, 0, b(0, n, k-1)):
seq(seq(T(n,k), k=0..n), n=0..12);
-
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, j], {j, 1, Min[t, u]}] + Sum[b[u + j - 1, o - j, j], {j, 1, Min[t, o]}]];
T[n_, k_] := b[0, n, k] - If[k == 0, 0, b[0, n, k - 1]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2018, 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
-
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);
-
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 *)
A288911
Number of permutations p of [n] such that 0p has a nonincreasing up-jump sequence.
Original entry on oeis.org
1, 1, 2, 5, 19, 80, 416, 2306, 14588, 98053, 724183, 5633793, 47416901, 417050215, 3914152702, 38288228393, 395496623939, 4241350801439, 47715403637219, 555476398869869, 6744406721447538, 84548532634924758, 1100301545470162305, 14751287346427752887
Offset: 0
-
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, t), j=1..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, t], {j, 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 *)
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 1-4 of 4 results.
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