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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.

%I A238121 #34 Sep 26 2023 11:34:19
%S A238121 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,
%T A238121 101,17,0,0,0,0,22,215,373,145,9,0,0,0,0,30,430,1174,836,146,4,0,0,0,
%U A238121 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
%N 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.
%C A238121 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.
%C A238121 Columns k=0-10 give: A000041, A241794, A241795, A241796, A241797, A241798, A241799, A241800, A241801, A241802, A241803.
%C A238121 T(2n,n) gives A241804.
%C A238121 T(2n+1,n) gives A241805.
%C A238121 Row sums are A000085.
%C A238121 T(n*(n+1)/2,n*(n-1)/2) = 1.
%C A238121 A238122 is another version with zeros omitted.
%H A238121 Joerg Arndt and Alois P. Heinz, <a href="/A238121/b238121.txt">Rows n = 0..50, flattened</a>
%e A238121 Triangle starts:
%e A238121     1;
%e A238121     1,    0;
%e A238121     2,    0,     0;
%e A238121     3,    1,     0,      0;
%e A238121     5,    5,     0,      0,      0;
%e A238121     7,   16,     3,      0,      0,      0;
%e A238121    11,   43,    21,      1,      0,      0,     0;
%e A238121    15,   99,   101,     17,      0,      0,     0,    0;
%e A238121    22,  215,   373,    145,      9,      0,     0,    0,  0;
%e A238121    30,  430,  1174,    836,    146,      4,     0,    0,  0, 0;
%e A238121    42,  834,  3337,   3846,   1324,    112,     1,    0,  0, 0, 0;
%e A238121    56, 1529,  8642,  15002,   8786,   1615,    66,    0,  0, 0, 0, 0;
%e A238121    77, 2765, 21148,  52132,  47013,  15403,  1582,   32,  0, 0, 0, 0, 0;
%e A238121   101, 4792, 48713, 164576, 214997, 112106, 21895, 1310, 14, 0, 0, 0, 0, 0;
%e A238121   ...
%e A238121 The T(5,1) = 16 ballot sequences of length n=5 with k=1 descent are (dots for zeros):
%e A238121 01:  [ . . . 1 . ]
%e A238121 02:  [ . . 1 . . ]
%e A238121 03:  [ . . 1 . 1 ]
%e A238121 04:  [ . . 1 . 2 ]
%e A238121 05:  [ . . 1 1 . ]
%e A238121 06:  [ . . 1 2 . ]
%e A238121 07:  [ . . 1 2 1 ]
%e A238121 08:  [ . 1 . . . ]
%e A238121 09:  [ . 1 . . 1 ]
%e A238121 10:  [ . 1 . . 2 ]
%e A238121 11:  [ . 1 . 1 2 ]
%e A238121 12:  [ . 1 . 2 3 ]
%e A238121 13:  [ . 1 2 . . ]
%e A238121 14:  [ . 1 2 . 1 ]
%e A238121 15:  [ . 1 2 . 3 ]
%e A238121 16:  [ . 1 2 3 . ]
%p A238121 b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
%p A238121       add(`if`(i=1 or l[i-1]>l[i], `if`(i<v, x, 1)*
%p A238121       b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
%p A238121       b(n-1, nops(l)+1, [l[], 1])))
%p A238121     end:
%p A238121 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n-1, 1, [1])):
%p A238121 seq(T(n), n=0..14);
%t A238121 b[n_, v_, l_] := b[n, v, l] = If[n<1, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i<v, x, 1]*b[n-1, i, ReplacePart[l, 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 *)
%K A238121 nonn,tabl
%O A238121 0,4
%A A238121 _Joerg Arndt_ and _Alois P. Heinz_, Feb 21 2014