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 A182172 Number A(n,k) of standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 44
 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 6, 1, 0, 1, 1, 2, 4, 9, 10, 1, 0, 1, 1, 2, 4, 10, 21, 20, 1, 0, 1, 1, 2, 4, 10, 25, 51, 35, 1, 0, 1, 1, 2, 4, 10, 26, 70, 127, 70, 1, 0, 1, 1, 2, 4, 10, 26, 75, 196, 323, 126, 1, 0, 1, 1, 2, 4, 10, 26, 76, 225, 588, 835, 252, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS Also the number A(n,k) of standard Young tableaux of n cells and <= k columns. A(n,k) is also the number of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,ak), where #(z,x) counts the letters x in word z.  The A(4,4) = 10 words of length 4 over alphabet {a,b,c,d} are: aaaa, aaab, aaba, abaa, aabb, abab, aabc, abac, abca, abcd. LINKS Alois P. Heinz, Antidiagonals n = 0..80, flattened Wikipedia, Young tableau FORMULA Conjecture: A(n,k) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * Product_{j=1..k} Gamma(j/2). - Vaclav Kotesovec, Sep 12 2013 EXAMPLE A(4,2) = 6, there are 6 standard Young tableaux of 4 cells and height <= 2:   +------+  +------+  +---------+  +---------+  +---------+  +------------+   | 1  3 |  | 1  2 |  | 1  3  4 |  | 1  2  4 |  | 1  2  3 |  | 1  2  3  4 |   | 2  4 |  | 3  4 |  | 2 .-----+  | 3 .-----+  | 4 .-----+  +------------+   +------+  +------+  +---+        +---+        +---+ Square array A(n,k) begins:   1,  1,  1,   1,   1,   1,   1,   1,   1, ...   0,  1,  1,   1,   1,   1,   1,   1,   1, ...   0,  1,  2,   2,   2,   2,   2,   2,   2, ...   0,  1,  3,   4,   4,   4,   4,   4,   4, ...   0,  1,  6,   9,  10,  10,  10,  10,  10, ...   0,  1, 10,  21,  25,  26,  26,  26,  26, ...   0,  1, 20,  51,  70,  75,  76,  76,  76, ...   0,  1, 35, 127, 196, 225, 231, 232, 232, ...   0,  1, 70, 323, 588, 715, 756, 763, 764, ... 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) option remember;       `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1\$n]),         g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))     end: A:= (n, k)-> g(n, k, []): seq(seq(A(n, d-n), n=0..d), d=0..15); MATHEMATICA h[l_List] := 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}]]; g[n_, i_, l_List] := 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]]]]]]; a[n_, k_] := g[n, k, {}]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 06 2013, translated from Maple *) CROSSREFS Columns k=0-12 give: A000007, A000012, A001405, A001006, A005817, A049401, A007579, A007578, A007580, A212915, A212916, A229053, A229068. Main diagonal gives A000085. A(2n,n) gives A293128. Cf. A047884, A049400, A226873, A240608. Sequence in context: A293285 A262553 A247506 * A143841 A276719 A276837 Adjacent sequences:  A182169 A182170 A182171 * A182173 A182174 A182175 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Apr 16 2012 STATUS approved

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Last modified July 1 18:15 EDT 2022. Contains 354973 sequences. (Running on oeis4.)