A241802 -id:A241802 - cp's OEIS frontend

A238121 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k descents, n>=0, 0<=k<=n.

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 5, 0, 0, 0, 7, 16, 3, 0, 0, 0, 11, 43, 21, 1, 0, 0, 0, 15, 99, 101, 17, 0, 0, 0, 0, 22, 215, 373, 145, 9, 0, 0, 0, 0, 30, 430, 1174, 836, 146, 4, 0, 0, 0, 0, 42, 834, 3337, 3846, 1324, 112, 1, 0, 0, 0, 0, 56, 1529, 8642, 15002, 8786, 1615, 66, 0, 0, 0, 0, 0, 77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 0, 0, 0, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Also number of standard Young tableaux such that there are k pairs of cells (v,v+1) with v+1 lying in a row above v.
Columns k=0-10 give: A000041, A241794, A241795, A241796, A241797, A241798, A241799, A241800, A241801, A241802, A241803.
T(2n,n) gives A241804.
T(2n+1,n) gives A241805.
Row sums are A000085.
T(n*(n+1)/2,n*(n-1)/2) = 1.
A238122 is another version with zeros omitted.

			Triangle starts:
    1;
    1,    0;
    2,    0,     0;
    3,    1,     0,      0;
    5,    5,     0,      0,      0;
    7,   16,     3,      0,      0,      0;
   11,   43,    21,      1,      0,      0,     0;
   15,   99,   101,     17,      0,      0,     0,    0;
   22,  215,   373,    145,      9,      0,     0,    0,  0;
   30,  430,  1174,    836,    146,      4,     0,    0,  0, 0;
   42,  834,  3337,   3846,   1324,    112,     1,    0,  0, 0, 0;
   56, 1529,  8642,  15002,   8786,   1615,    66,    0,  0, 0, 0, 0;
   77, 2765, 21148,  52132,  47013,  15403,  1582,   32,  0, 0, 0, 0, 0;
  101, 4792, 48713, 164576, 214997, 112106, 21895, 1310, 14, 0, 0, 0, 0, 0;
  ...
The T(5,1) = 16 ballot sequences of length n=5 with k=1 descent are (dots for zeros):
01:  [ . . . 1 . ]
02:  [ . . 1 . . ]
03:  [ . . 1 . 1 ]
04:  [ . . 1 . 2 ]
05:  [ . . 1 1 . ]
06:  [ . . 1 2 . ]
07:  [ . . 1 2 1 ]
08:  [ . 1 . . . ]
09:  [ . 1 . . 1 ]
10:  [ . 1 . . 2 ]
11:  [ . 1 . 1 2 ]
12:  [ . 1 . 2 3 ]
13:  [ . 1 2 . . ]
14:  [ . 1 2 . 1 ]
15:  [ . 1 2 . 3 ]
16:  [ . 1 2 3 . ]

Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened

b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
      add(`if`(i=1 or l[i-1]>l[i], `if`(i (p-> seq(coeff(p, x, i), i=0..n))(b(n-1, 1, [1])):
seq(T(n), n=0..14);

b[n_, v_, l_] := b[n, v, l] = If[n<1, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *)

A238122 Irregular triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k descents, n>=0, 0<=k<=A083920(n-1).

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 5, 7, 16, 3, 11, 43, 21, 1, 15, 99, 101, 17, 22, 215, 373, 145, 9, 30, 430, 1174, 836, 146, 4, 42, 834, 3337, 3846, 1324, 112, 1, 56, 1529, 8642, 15002, 8786, 1615, 66, 77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 101, 4792, 48713, 164576, 214997, 112106, 21895, 1310, 14

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Same as A238121, with zeros omitted.
Columns k=0-10 give: A000041, A241794, A241795, A241796, A241797, A241798, A241799, A241800, A241801, A241802, A241803.
T(2n,n) gives A241804.
T(2n+1,n) gives A241805.
Row sums are A000085.
T(n*(n+1)/2,n*(n-1)/2) = 1.

			T(5,0) = 7: [1,1,1,1,1], [1,1,1,1,2], [1,1,1,2,2], [1,1,1,2,3], [1,1,2,2,3], [1,1,2,3,4], [1,2,3,4,5].
T(5,1) = 16: [1,1,1,2,1], [1,1,2,1,1], [1,1,2,1,2], [1,1,2,1,3], [1,1,2,2,1], [1,1,2,3,1], [1,1,2,3,2], [1,2,1,1,1], [1,2,1,1,2], [1,2,1,1,3], [1,2,1,2,3], [1,2,1,3,4], [1,2,3,1,1], [1,2,3,1,2], [1,2,3,1,4], [1,2,3,4,1].
T(5,2) = 3: [1,2,1,2,1], [1,2,1,3,1], [1,2,1,3,2].
Triangle starts:
00:   1;
01:   1;
02:   2;
03:   3,    1;
04:   5,    5;
05:   7,   16,      3;
06:  11,   43,     21,      1;
07:  15,   99,    101,     17;
08:  22,  215,    373,    145,      9;
09:  30,  430,   1174,    836,    146,      4;
10:  42,  834,   3337,   3846,   1324,    112,      1;
11:  56, 1529,   8642,  15002,   8786,   1615,     66;
12:  77, 2765,  21148,  52132,  47013,  15403,   1582,    32;
13: 101, 4792,  48713, 164576, 214997, 112106,  21895,  1310,  14;
14: 135, 8216, 108147, 484609, 874413, 672015, 215849, 26159, 932, 5;
...

Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened

b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
      add(`if`(i=1 or l[i-1]>l[i], `if`(i(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 1, [1])):
seq(T(n), n=0..14);

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i l[[i]]+1]], 0], {i, 1, Length[ l ]}] + b[n-1, Length[l]+1, Append[l, 1]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)

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A238121 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k descents, n>=0, 0<=k<=n.

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