A229066 -id:A229066 - cp's OEIS frontend

A229892 Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 5, 3, 1, 1, 0, 16, 6, 4, 1, 1, 0, 61, 26, 10, 5, 1, 1, 0, 272, 71, 20, 15, 6, 1, 1, 0, 1385, 413, 125, 35, 21, 7, 1, 1, 0, 7936, 1456, 461, 70, 56, 28, 8, 1, 1, 0, 50521, 10576, 1301, 574, 126, 84, 36, 9, 1, 1

Offset: 0

Alois P. Heinz, Oct 02 2013

T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. T(n,k) = T(n,n) = A000012(n) = 1 for k>n.
T(2*n,n) = C(2*n-1,n) = A088218(n) = A001700(n-1) for n>0.
T(2*n+1,n) = C(2*n,n) = A000984(n).
T(2*n+1,n+1) = C(2n,n-1) = A001791(n) for n>0.

			Triangle T(n,k) begins:
  1;
  1,    1;
  0,    1,   1;
  0,    2,   1,   1;
  0,    5,   3,   1,  1;
  0,   16,   6,   4,  1,  1;
  0,   61,  26,  10,  5,  1, 1;
  0,  272,  71,  20, 15,  6, 1, 1;
  0, 1385, 413, 125, 35, 21, 7, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=1-10 give: A000111, A058258, A229884, A229885, A229886, A229887, A229888, A229889, A229890, A229891.

Cf. A227941, A229066, A229551.

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=k,
       b(o-j, u+j-1, 1, k), b(u+j-1, o-j, t+1, k)), j=1..o))
    end:
T:= (n, k)-> `if`(k+1>=n, 1, `if`(k=0, 0, b(0, n, 0, k))):
seq(seq(T(n, k), k=0..n), n=0..10);

b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, Sum[If[t == k, b[o-j, u+j-1, 1, k], b[u+j-1, o-j, t+1, k]], {j, 1, o}]]; t[n_, k_] := If[k+1 >= n, 1, If[k == 0, 0, b[0, n, 0, k]]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

T(7,3) = 20: 1237654, 1247653, 1257643, 1267543, 1347652, 1357642, 1367542, 1457632, 1467532, 1567432, 2347651, 2357641, 2367541, 2457631, 2467531, 2567431, 3457621, 3467521, 3567421, 4567321.

A229551 Number of 1 up, 1 down, 2 up, 2 down, ... permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 40, 111, 643, 2261, 6176, 53560, 265001, 976535, 10699235, 65839306, 297528021, 1096638993, 16254932942, 131192702293, 760059358527, 3527632148650, 63700463354263, 620906514026512, 4309068955961383, 23776534616426566, 110660256825406666

Offset: 0

Alois P. Heinz, Oct 02 2013

			a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 9: 13245, 14235, 15234, 23145, 24135, 25134, 34125, 35124, 45123.
a(6) = 40: 132465, 132564, ..., 561342, 562341.
a(7) = 111: 1324765, 1325764, ..., 6724531, 6734521.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A227941, A229066, A229892, A247550.

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, `if`(t>0,
       add(b(u+j-1, o-j, `if`(t=k, [-1, k], [t+1, k])[]), j=1..o),
       add(b(u-j, o+j-1, `if`(t=-k, [1, k+1], [t-1, k])[]), j=1..u)))
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 1, 1), j=1..n)):
seq(a(n), n=0..35);

b[u_, o_, t_, k_] := b[u, o, t, k] = If[u + o == 0, 1, If[t > 0, Sum[b[u + j - 1, o - j, If[t == k, -1, t + 1], k], {j, 1, o}], Sum[b[u - j, o + j - 1, If[t == -k, 1, t - 1], If[t == -k, k + 1, k]], {j, 1, u}]]];
a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, 1, 1], {j, 1, n}]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 16 2018, after Alois P. Heinz *)

A227941 Number of 1 up, 3 down, 5 up, 7 down, ... permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 19, 55, 125, 245, 434, 4060, 21186, 81212, 254813, 692678, 1688555, 3776432, 60101767, 511650887, 3089821383, 14824989723, 60057570858, 213302293918, 681247718668, 1992449334436, 5409214694961, 132273848506202, 1692162553490943

Offset: 0

Alois P. Heinz, Oct 03 2013

			a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 3: 1432, 2431, 3421.
a(5) = 4: 15432, 25431, 35421, 45321.
a(6) = 19: 154326, 164325, 165324, 165423, 254316, 264315, 265314, 265413, 354216, 364215, 365214, 365412, 453216, 463215, 465213, 465312, 563214, 564213, 564312.
a(7) = 55: 1543267, 1643257, ..., 6753124, 6754123.
a(8) = 125: 15432678, 16432578, ..., 78641235, 78651234.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A005408, A229066, A229551, A229892.

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=2*k-1,
       b(o-j, u+j-1, 1, k+1), b(u+j-1, o-j, t+1, k)), j=1..o))
    end:
a:= n-> b(0, n, 0, 1):
seq(a(n), n=0..35);

b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, Sum[If[t == 2*k-1, b[o-j, u+j-1, 1, k+1], b[u+j-1, o-j, t+1, k]], {j, 1, o}]];
a[n_] := b[0, n, 0, 1];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Sep 01 2022, after Alois P. Heinz *)

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A229892 Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A229551 Number of 1 up, 1 down, 2 up, 2 down, ... permutations of [n].

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A227941 Number of 1 up, 3 down, 5 up, 7 down, ... permutations of [n].

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