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 A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k. 5
 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 5, 2, 1, 1, 6, 10, 9, 5, 2, 1, 1, 7, 14, 16, 10, 5, 2, 1, 1, 8, 19, 24, 19, 11, 5, 2, 1, 1, 9, 24, 37, 32, 21, 11, 5, 2, 1, 1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1, 1, 11, 37, 71, 79, 66, 41, 23, 11, 5, 2, 1, 1, 12, 44, 93, 117, 106, 74, 43, 23, 11, 5, 2, 1, 1, 13, 52, 122, 166, 166, 125, 80, 44, 23, 11, 5, 2, 1, 1, 14, 61, 153, 231, 251, 204, 139, 83, 45, 23, 11, 5, 2, 1, 1, 15, 70, 193, 311, 367, 322, 236, 147, 85, 45, 23, 11, 5, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums equal A003293. Reverse of rows seem to converge to A005986: 1, 2, 5, 11, 23, 45, 87, 160, ... LINKS EXAMPLE T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are 114 , 123 , 222 , 11 , 12 , 13 , 1 4 3 2 2 3 Triangle starts: 1; 1, 1; 1, 2, 1; 1, 3, 2, 1; 1, 4, 4, 2, 1; 1, 5, 7, 5, 2, 1; 1, 6, 10, 9, 5, 2, 1; 1, 7, 14, 16, 10, 5, 2, 1; 1, 8, 19, 24, 19, 11, 5, 2, 1; 1, 9, 24, 37, 32, 21, 11, 5, 2, 1; 1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1; MATHEMATICA hooklength[(par_)?PartitionQ]:=Table[Count[par, q_ /; q>=j] +1-i +par[[i]] -j, {i, Length[par]}, {j, par[[i]]} ]; Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1&) /@ Flatten[hooklength[#1]], {q, 0, w}]&) /@ Partitions[n]], {w, 24}, {n, w}] CROSSREFS Cf. A003293, A005986. Sequence in context: A092905 A052509 A172119 * A227588 A093628 A186807 Adjacent sequences: A228122 A228123 A228124 * A228126 A228127 A228128 KEYWORD nonn,tabl AUTHOR Wouter Meeussen, Aug 11 2013 STATUS approved

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Last modified January 31 18:34 EST 2023. Contains 359980 sequences. (Running on oeis4.)