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 A210391 Number A(n,k) of semistandard Young tableaux over all partitions of n with maximal element <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 21
 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 6, 1, 0, 1, 5, 16, 19, 9, 1, 0, 1, 6, 25, 44, 39, 12, 1, 0, 1, 7, 36, 85, 116, 69, 16, 1, 0, 1, 8, 49, 146, 275, 260, 119, 20, 1, 0, 1, 9, 64, 231, 561, 751, 560, 189, 25, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux Wikipedia, Young tableau FORMULA G.f. of column k: 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)). A(n,k) = Sum_{i=0..k} C(k,i) * A138177(n,k-i). - Alois P. Heinz, Apr 06 2015 EXAMPLE Square array A(n,k) begins:   1,  1,   1,   1,   1,    1,    1, ...   0,  1,   2,   3,   4,    5,    6, ...   0,  1,   4,   9,  16,   25,   36, ...   0,  1,   6,  19,  44,   85,  146, ...   0,  1,   9,  39, 116,  275,  561, ...   0,  1,  12,  69, 260,  751, 1812, ...   0,  1,  16, 119, 560, 1955, 5552, ... MAPLE # First program: h:= (l, k)-> mul(mul((k+j-i)/(1+l[i] -j +add(`if`(l[t]>=j, 1, 0)                  , t=i+1..nops(l))), j=1..l[i]), i=1..nops(l)): g:= proc(n, i, k, l)       `if`(n=0, h(l, k), `if`(i<1, 0, g(n, i-1, k, l)+       `if`(i>n, 0, g(n-i, i, k, [l[], i]))))     end: A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])): seq(seq(A(n, d-n), n=0..d), d=0..12); # second program: gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)): A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA (* First program: *) h[l_, k_] := Product[Product[(k+j-i)/(1+l[[i]]-j + Sum[If[l[[t]] >= j, 1, 0], {t, i+1, Length[l]}]), {j, 1, l[[i]]}], {i, 1, Length[l]}]; g [n_, i_, k_, l_] := If[n == 0, h[l, k], If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a[n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* second program: *) gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); a[n_, k_] := Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *) CROSSREFS Rows n=0-10 give: A000012, A001477, A000290, A005900, A139594, A210427, A210428, A210429, A210430, A210431, A210432. Columns k=0-8 give: A000007, A000012, A002620(n+2), A038163, A054498, A181477, A181478, A181479, A181480. Main diagonal gives: A209673. Cf. A138177, A191714. Sequence in context: A323224 A118340 A213276 * A071921 A003992 A246118 Adjacent sequences:  A210388 A210389 A210390 * A210392 A210393 A210394 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 20 2012 STATUS approved

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Last modified May 28 17:28 EDT 2020. Contains 334684 sequences. (Running on oeis4.)