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A238123 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k largest parts, n >= k >= 0.

%I A238123 #43 Jun 19 2018 05:56:45
%S A238123 1,0,1,0,1,1,0,3,0,1,0,7,2,0,1,0,20,5,0,0,1,0,56,14,5,0,0,1,0,182,35,
%T A238123 14,0,0,0,1,0,589,132,28,14,0,0,0,1,0,2088,399,90,42,0,0,0,0,1,0,7522,
%U A238123 1556,285,90,42,0,0,0,0,1,0,28820,5346,1232,165,132,0,0,0,0,0,1
%N A238123 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k largest parts, n >= k >= 0.
%C A238123 Also number of standard Young tableaux with last row of length k.
%H A238123 Joerg Arndt and Alois P. Heinz, <a href="/A238123/b238123.txt">Rows n = 0..60, flattened</a>
%H A238123 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A238123 Triangle starts:
%e A238123 00: 1;
%e A238123 01: 0,      1;
%e A238123 02: 0,      1,     1;
%e A238123 03, 0,      3,     0,     1;
%e A238123 04: 0,      7,     2,     0,    1;
%e A238123 05: 0,     20,     5,     0,    0,   1;
%e A238123 06: 0,     56,    14,     5,    0,   0,   1;
%e A238123 07: 0,    182,    35,    14,    0,   0,   0, 1;
%e A238123 08: 0,    589,   132,    28,   14,   0,   0, 0, 1;
%e A238123 09: 0,   2088,   399,    90,   42,   0,   0, 0, 0, 1;
%e A238123 10: 0,   7522,  1556,   285,   90,  42,   0, 0, 0, 0, 1;
%e A238123 11: 0,  28820,  5346,  1232,  165, 132,   0, 0, 0, 0, 0, 1;
%e A238123 12: 0, 113092, 21515,  4378,  737, 297, 132, 0, 0, 0, 0, 0, 1;
%e A238123 13: 0, 464477, 82940, 17082, 3003, 572, 429, 0, 0, 0, 0, 0, 0, 1;
%e A238123 ...
%e A238123 The T(6,2)=14 ballot sequences of length 6 with 2 maximal elements are (dots for zeros):
%e A238123 01:  [ . . . . 1 1 ]
%e A238123 02:  [ . . . 1 . 1 ]
%e A238123 03:  [ . . . 1 1 . ]
%e A238123 04:  [ . . 1 . . 1 ]
%e A238123 05:  [ . . 1 . 1 . ]
%e A238123 06:  [ . . 1 1 . . ]
%e A238123 07:  [ . . 1 1 2 2 ]
%e A238123 08:  [ . . 1 2 1 2 ]
%e A238123 09:  [ . 1 . . . 1 ]
%e A238123 10:  [ . 1 . . 1 . ]
%e A238123 11:  [ . 1 . 1 . . ]
%e A238123 12:  [ . 1 . 1 2 2 ]
%e A238123 13:  [ . 1 . 2 1 2 ]
%e A238123 14:  [ . 1 2 . 1 2 ]
%e A238123 The T(8,4)=14 such ballot sequences of length 8 and 4 maximal elements are:
%e A238123 01:  [ . . . . 1 1 1 1 ]
%e A238123 02:  [ . . . 1 . 1 1 1 ]
%e A238123 03:  [ . . . 1 1 . 1 1 ]
%e A238123 04:  [ . . . 1 1 1 . 1 ]
%e A238123 05:  [ . . 1 . . 1 1 1 ]
%e A238123 06:  [ . . 1 . 1 . 1 1 ]
%e A238123 07:  [ . . 1 . 1 1 . 1 ]
%e A238123 08:  [ . . 1 1 . . 1 1 ]
%e A238123 09:  [ . . 1 1 . 1 . 1 ]
%e A238123 10:  [ . 1 . . . 1 1 1 ]
%e A238123 11:  [ . 1 . . 1 . 1 1 ]
%e A238123 12:  [ . 1 . . 1 1 . 1 ]
%e A238123 13:  [ . 1 . 1 . . 1 1 ]
%e A238123 14:  [ . 1 . 1 . 1 . 1 ]
%e A238123 These are the (reversed) Dyck words of semi-length 4.
%p A238123 b:= proc(n, l) option remember; `if`(n<1, x^l[-1],
%p A238123       b(n-1, [l[], 1]) +add(`if`(i=1 or l[i-1]>l[i],
%p A238123       b(n-1, subsop(i=l[i]+1, l)), 0), i=1..nops(l)))
%p A238123     end:
%p A238123 T:= n->`if`(n=0, 1, (p->seq(coeff(p, x, i), i=0..n))(b(n-1, [1]))):
%p A238123 seq(T(n), n=0..12);
%p A238123 # second Maple program (counting SYT):
%p A238123 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A238123        add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A238123     end:
%p A238123 g:= proc(n, i, l) `if`(n=0 or i=1, h([l[], 1$n])*x^`if`(n>0, 1,
%p A238123        `if`(l=[], 0, l[-1])), g(n, i-1, l)+
%p A238123        `if`(i>n, 0, g(n-i, i, [l[], i])))
%p A238123     end:
%p A238123 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n, n, [])):
%p A238123 seq(T(n), n=0..12);
%t A238123 b[n_, l_List] :=  b[n, l] = If[n<1, x^l[[-1]], b[n-1, Append[l, 1]] +  Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; T[n_] := If[n == 0, 1, Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, {1}]]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 07 2015, translated from Maple *)
%o A238123 (PARI) (A238123(n,k)=if(k, vecsum(apply(p->n!/Hook(Vecrev(p)), select(p->p[1]==k,partitions(n,[k,n])))), !n)); Hook(P,h=vector(P[1]),L=P[#P])={prod(i=1, L, h[i]=L-i+1)*prod(i=1,#P-1, my(D=-L+L=P[#P-i]); prod(k=0,L-1,h[L-k]+=min(k,D)+1))} \\  _M. F. Hasler_, Jun 03 2018
%Y A238123 The terms T(2*n,n) are the Catalan numbers (A000108).
%Y A238123 Columns k=0-10 give: A000007, A238124, A244099, A244100, A244101, A244102, A244103, A244104, A244105, A244106, A244107.
%Y A238123 Row sums give A000085.
%Y A238123 Cf. A026794.
%K A238123 nonn,tabl
%O A238123 0,8
%A A238123 _Joerg Arndt_ and _Alois P. Heinz_, Feb 21 2014