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%I #38 Oct 29 2021 05:06:06
%S 1,1,1,2,2,1,4,4,3,1,10,10,9,4,1,26,26,25,16,5,1,76,76,75,56,25,6,1,
%T 232,232,231,197,105,36,7,1,764,764,763,694,441,176,49,8,1,2620,2620,
%U 2619,2494,1785,856,273,64,9,1,9496,9496,9495,9244,7308,3952,1506,400,81,10,1
%N Number T(n,k) of standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= k. T(4,3) = 4: 1234, 1243, 1324, 2134; T(3,0) = T(3,1) = 4: 123, 132, 213, 321; T(5,3) = 16: 12345, 12354, 12435, 12543, 13245, 13254, 14325, 14523, 15342, 21345, 21354, 21435, 32145, 34125, 42315, 52341.
%H Alois P. Heinz, <a href="/A182222/b182222.txt">Rows n = 0..50, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Involution_(mathematics)">Involution (mathematics)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F T(n,k) = A182172(n,n) - A182172(n,k-1) for k>0, T(n,0) = A182172(n,n).
%e T(4,3) = 4, there are 4 standard Young tableaux of 4 cells and height >= 3:
%e +---+ +------+ +------+ +------+
%e | 1 | | 1 2 | | 1 3 | | 1 4 |
%e | 2 | | 3 .--+ | 2 .--+ | 2 .--+
%e | 3 | | 4 | | 4 | | 3 |
%e | 4 | +---+ +---+ +---+
%e +---+
%e Triangle T(n,k) begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 4, 4, 3, 1;
%e 10, 10, 9, 4, 1;
%e 26, 26, 25, 16, 5, 1;
%e 76, 76, 75, 56, 25, 6, 1;
%e 232, 232, 231, 197, 105, 36, 7, 1;
%e 764, 764, 763, 694, 441, 176, 49, 8, 1;
%e ...
%p h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
%p add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p end:
%p g:= proc(n, i, l) option remember;
%p `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
%p g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p end:
%p T:= (n, k)-> g(n, n, []) -`if`(k=0, 0, g(n, k-1, [])):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t h[l_] := Module[{n = Length[l]}, Sum[i, {i, l}]! / Product[ Product[1 + l[[i]] - j + Sum [If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
%t g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
%t t[n_, k_] := g[n, n, {}] - If[k == 0, 0, g[n, k-1, {}]];
%t Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 12 2013, translated from Maple *)
%Y Columns 0-10 give: A000085, A000085 (for n>0), A001189, A218263, A218264, A218265, A218266, A218267, A218268, A218269, A218262.
%Y Diagonal and lower diagonals give: A000012, A000027(n+1), A000290(n+1) for n>0, A131423(n+1) for n>1.
%Y T(2n,n) gives A318289.
%Y Cf. A047884, A049400, A182172.
%K nonn,tabl
%O 0,4
%A _Alois P. Heinz_, Apr 19 2012
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