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 A117506 Irregular triangle read by rows: dimensions of the irreducible representations of the symmetric group S_n. 26
 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 6, 5, 4, 1, 1, 5, 9, 5, 10, 16, 5, 10, 9, 5, 1, 1, 6, 14, 14, 15, 35, 21, 21, 20, 35, 14, 15, 14, 6, 1, 1, 7, 20, 28, 14, 21, 64, 70, 56, 42, 35, 90, 56, 70, 14, 35, 64, 28, 21, 20, 7, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The n-th row has partition(n) = A000041(n) entries. Also the numbers of standard Young tableaux for Young diagrams (or partitions). Also "generalized" Catalan numbers. For a partition of n, n=(n_1+...+n_d), this is the number of integral lattice paths from (0,...,0) to (n_1,...,n_d) such that for any point p=(p_1,...p_d) on such a path p_i is never less than p_j whenever i (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),              k=i+1..n), j=1..l[i]), i=1..n))(nops(l)): g:= (n, i, l)-> `if`(n=0 or i=1, [h([l[], 1\$n])],     [g(n, i-1, l)[], g(n-i, min(n-i, i), [l[], i])[]]): T:= n-> map(x-> n!/x, g(n\$2, []))[]: seq(T(n), n=0..10);  # Alois P. Heinz, Nov 05 2015 MATHEMATICA h[l_List] := Function[n, Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]][Length[l]]; g[n_, i_, l_List] := If[n==0 || i==1, Join[{h[Join[l, Array[1&, n]]]}], If[i<1, {}, Join[{g[n, i-1, l]}, If[i>n, {}, g[n-i, i, Join[l, {i}]]]]]] // Flatten; T[n_] := n!/ g[n, n, {}]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *) CROSSREFS Cf. A000041, A000085 (row sums), A060240 (rows sorted), A263003. Sequence in context: A006843 A324797 A049456 * A179205 A055089 A060117 Adjacent sequences:  A117503 A117504 A117505 * A117507 A117508 A117509 KEYWORD nonn,easy,tabf AUTHOR Wolfdieter Lang, Apr 13 2006 EXTENSIONS Row n=0 prepended by Alois P. Heinz, Nov 05 2015 STATUS approved

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Last modified October 14 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)