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 A047884 Triangle of numbers a(n,k) = number of Young tableaux with n cells and k rows (1 <= k <= n); also number of self-inverse permutations on n letters in which the length of the longest scattered (i.e., not necessarily contiguous) increasing subsequence is k. 18
 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 9, 11, 4, 1, 1, 19, 31, 19, 5, 1, 1, 34, 92, 69, 29, 6, 1, 1, 69, 253, 265, 127, 41, 7, 1, 1, 125, 709, 929, 583, 209, 55, 8, 1, 1, 251, 1936, 3356, 2446, 1106, 319, 71, 9, 1, 1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 REFERENCES W. Fulton, Young Tableaux, Cambridge, 1997. D. Stanton and D. White, Constructive Combinatorics, Springer, 1986. LINKS Alois P. Heinz, Rows n = 1..68, flattened R. P. Stanley, A combinatorial miscellany EXAMPLE For n=3 the 4 tableaux are 1 2 3 . 1 2 . 1 3 . 1 . . . . 3 . . 2 . . 2 . . . . . . . . . . 3 Triangle begins:   1;   1,   1;   1,   2,    1;   1,   5,    3,     1;   1,   9,   11,     4,     1;   1,  19,   31,    19,     5,    1;   1,  34,   92,    69,    29,    6,    1;   1,  69,  253,   265,   127,   41,    7,   1;   1, 125,  709,   929,   583,  209,   55,   8,  1;   1, 251, 1936,  3356,  2446, 1106,  319,  71,  9,  1;   1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10,  1; MAPLE h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+        add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)     end: g:= proc(n, i, l) `if`(n=0 or i=1, (p->h(p)*x^`if`(p=[], 0, p[1]))       ([l[], 1\$n]), add(g(n-i*j, i-1, [l[], i\$j]), j=0..n/i))     end: T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n\$2, [])): seq(T(n), n=1..14); # Alois P. Heinz, Apr 16 2012, revised Mar 05 2014 MATHEMATICA Table[ Plus@@( NumberOfTableaux/@ Reverse/@Union[ Sort/@(Compositions[ n-m, m ]+1) ]), {n, 12}, {m, n} ] h[l_] := With[{n=Length[l]}, Total[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}]]; g[n_, i_, l_] := If[n== 0|| i==1, Function[p, h[p]*x^If[p == {}, 0, p[[1]] ] ] [ Join[l, Array[1&, n]]], Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n, {}]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *) CROSSREFS Row sums give A000085. Cf. A049400, A049401, and A178249 which imposes contiguity. Columns k=1-10 give: A000012, A014495, A217323, A217324, A217325, A217326, A217327, A217328, A217321, A217322. - Alois P. Heinz, Oct 03 2012 a(2n,n) gives A267436. Sequence in context: A107735 A137570 A079213 * A124328 A055818 A106240 Adjacent sequences:  A047881 A047882 A047883 * A047885 A047886 A047887 KEYWORD nonn,tabl,nice,easy AUTHOR EXTENSIONS Definition amended ('scattered' added) by Wouter Meeussen, Dec 22 2010 STATUS approved

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Last modified February 23 23:01 EST 2020. Contains 332195 sequences. (Running on oeis4.)