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%I #23 Aug 27 2021 02:28:50
%S 1,0,1,0,0,2,0,0,1,3,0,0,1,2,7,0,0,1,3,8,14,0,0,1,4,11,19,41,0,0,1,7,
%T 19,34,64,107,0,0,1,11,32,62,119,202,337,0,0,1,21,64,131,248,418,671,
%U 1066,0,0,1,36,124,277,545,943,1518,2361,3691
%N Number T(n,k) of standard Young tableaux with n cells where k is the largest value in the last row; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C T(0,0) = 1 by convention.
%C Also the number of ballot sequences of length n having the last occurrence of the maximal value at position k.
%C T(n,3) = A051920(n-3) for n>3.
%C T(2n,n) gives A246818.
%C Main diagonal gives A238728.
%C Row sums give A000085.
%H Joerg Arndt and Alois P. Heinz, <a href="/A238727/b238727.txt">Rows n = 0..43, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e The 10 tableaux with n=4 cells sorted by largest value in the last row:
%e :[1 3 4]:[1 4] [1 2 4]:[1] [1 2] [1 3] [1 2 3] [1 2] [1 3] [1 2 3 4]:
%e :[2] :[2] [3] :[2] [3] [2] [4] [3 4] [2 4] :
%e : :[3] :[3] [4] [4] :
%e : : :[4] :
%e : --2-- : -----3----- : ---------------------4--------------------- :
%e The 10 ballot sequences of length 4 sorted by the position of the last occurrence of the maximal value:
%e [1, 2, 1, 1] -> 2 } -- 1
%e [1, 2, 3, 1] -> 3 \ __ 2
%e [1, 1, 2, 1] -> 3 /
%e [1, 2, 3, 4] -> 4 \
%e [1, 1, 2, 3] -> 4 \
%e [1, 2, 1, 3] -> 4 \
%e [1, 1, 1, 2] -> 4 } 7
%e [1, 1, 2, 2] -> 4 /
%e [1, 2, 1, 2] -> 4 /
%e [1, 1, 1, 1] -> 4 /
%e thus row 4 = [0, 0, 1, 2, 7].
%e Triangle T(n,k) begins:
%e 00: 1;
%e 01: 0, 1;
%e 02: 0, 0, 2;
%e 03: 0, 0, 1, 3;
%e 04: 0, 0, 1, 2, 7;
%e 05: 0, 0, 1, 3, 8, 14;
%e 06: 0, 0, 1, 4, 11, 19, 41;
%e 07: 0, 0, 1, 7, 19, 34, 64, 107;
%e 08: 0, 0, 1, 11, 32, 62, 119, 202, 337;
%e 09: 0, 0, 1, 21, 64, 131, 248, 418, 671, 1066;
%e 10: 0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691;
%p h:= proc(l) option remember; local n, s; n:= nops(l); s:= add(i, i=l);
%p `if`(n=0, 1, add(`if`(i<n and l[i]>l[i+1], h(subsop(i=l[i]-1, l)),
%p `if`(i=n, (p->add(coeff(p,x,j)*x^`if`(j<s, s, j), j=0..degree(p)))
%p (h(subsop(i=`if`(l[i]>1, l[i]-1, [][]), l))), 0)), i=1..n))
%p end:
%p g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]),
%p add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
%p T:= n-> (p->seq(coeff(p, x, i), i=0..n))(g(n$2, [])):
%p seq(T(n), n=0..12);
%t h[l_] := h[l] = With[{n = Length[l], s = Total[l]},
%t If[n == 0, 1, Sum[If[i < n && l[[i]] > l[[i + 1]],
%t h[ReplacePart[l, i -> l[[i]] - 1]], If[i == n, Function[p,
%t Sum[Coefficient[p, x, j] x^If[j < s, s, j], {j, 0,
%t Exponent[p, x]}]][h[ReplacePart[l, i -> If[l[[i]] > 1,
%t l[[i]] - 1, Nothing]]]], 0]], {i, n}]]];
%t g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]],
%t Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
%t T[n_] := CoefficientList[g[n, n, {}], x];
%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Aug 27 2021, after Maple code )*
%K nonn,tabl
%O 0,6
%A _Joerg Arndt_ and _Alois P. Heinz_, Mar 03 2014
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