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 A228117 Number of partitions of n that have hookset {1,2,...,k} for some k. 0
 1, 1, 2, 2, 3, 4, 4, 6, 7, 9, 10, 16, 14, 23, 24, 33, 33, 50, 50, 71, 75, 101, 103, 146, 151, 201, 211, 280, 292, 389, 409, 519, 573, 707, 765, 960, 1043, 1276, 1393, 1704, 1870, 2258, 2483, 2970, 3281, 3920, 4290, 5101, 5659, 6640, 7318, 8628, 9506, 11081 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS It appears to be the case that the difference between entry a(2n-1) and a(2n) is substantially less than the difference between a(2n) and a(2n+1), after a few initial exceptions. LINKS EXAMPLE a(7) = 6, counting the partitions (7), (43), (331), (322), (2221), and (111111).  The hooklengths of (7) are {1,2,3,4,5,6,7}, and the hooklengths of (322) are {1,1,2,2,3,4,5}. MAPLE h:= proc(l) local n, s; n:=nops(l); s:= {seq(seq(1+l[i]-j        +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)};        `if`(s={\$1..max(s[], 0)}, 1, 0)     end: g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1\$n]), `if`(i<1, 0,              g(n, i-1, l)+`if`(i>n, 0, g(n-i, i, [l[], i])))): a:= n-> g(n\$2, []): seq(a(n), n=0..30);  # Alois P. Heinz, Aug 12 2013 MATHEMATICA << "Combinatorica`" HookSet[Lambda_] := Module[{i, j, k, HookHolder},   HookHolder = {};   HS = {};   For[i = 1, i < Length[Lambda] + 1, i++,    For[j = 1, j < Lambda[[i]] + 1, j++,     CurrentHook =      Lambda[[i]] - j + TransposePartition[Lambda][[j]] - i + 1;     If[! MemberQ[HS, CurrentHook],      HookHolder = Append[HS, CurrentHook]; HS = HookHolder]     ]    ];   HookHolder = Sort[HS];   HS = HookHolder;   Return[HS]] For[i = 1, i < 31, i++, For[j = 1, j < PartitionsP[i] + 1, j++,   CurrSet=HookSet[Partitions[i][[j]]];   If[CurrSet == Table[i, {i, 1, Length[CurrSet]}],    SGFHolder = SegGenFn + q^i;    SegGenFn = SGFHolder]   ] ] (* second program: *) h[l_] := Module[{n, s}, n = Length[l]; s = Table[Table[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] // Flatten // Union; If[s == Range[Max[Append[s, 0]]], 1, 0]]; g[n_, i_, l_] := g[n, i, l] = If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i<1, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i, Append[l, i]]]]]; a[n_] := g[n, n, {}]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *) CROSSREFS Cf. A158291, the number of partitions which have hookset {1,2,...,n}, not counting multiplicities. Sequence in context: A143038 A029040 A053281 * A286218 A094997 A173673 Adjacent sequences:  A228114 A228115 A228116 * A228118 A228119 A228120 KEYWORD nonn AUTHOR William J. Keith, Aug 10 2013 EXTENSIONS a(31)-a(53) from Alois P. Heinz, Aug 12 2013 STATUS approved

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Last modified April 6 15:41 EDT 2020. Contains 333276 sequences. (Running on oeis4.)