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.
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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:
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);
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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 *)
A291688
Number of permutations p of [2n] such that 0p has a nonincreasing jump sequence beginning with n.
Original entry on oeis.org
1, 1, 5, 36, 327, 3392, 38795, 469662, 5935728, 77416352, 1035050705, 14094000938, 195075365778, 2734475097609, 38747262233793, 554199475506095, 7990492729051526, 115995691148658656, 1694340616136589743, 24882428969673439384, 367160435328847044586
Offset: 0
a(2) = 5: 2134, 2314, 2341, 2413, 2431.
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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, 2*n, n)-`if`(n=0, 0, b(0, 2*n, n-1)):
seq(a(n), n=0..25);
-
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, 2n, n] - If[n == 0, 0, b[0, 2n, n - 1]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)
A303203
Number of permutations p of [2n+1] such that 0p has a nonincreasing jump sequence beginning with n+1.
Original entry on oeis.org
1, 2, 12, 98, 988, 10872, 129520, 1609176, 20786804, 274792342, 3719296036, 51119572738, 712918642042, 10055531355652, 143287150725298, 2058368140071146, 29796314629826814, 434051906728752164, 6359063811707227298, 93635249115751389952, 1385028309353547034876
Offset: 0
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
end:
a:= n-> b(0, 2*n+1, n+1)-b(0, 2*n+1, n):
seq(a(n), n=0..25);
-
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]}]];
a[n_] := b[0, 2n+1, n+1] - b[0, 2n+1, n];
a /@ Range[0, 25] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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