A228117 Number of partitions of n that have hookset {1,2,...,k} for some k.

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

Offset: 0

Author

William J. Keith, Aug 10 2013

Keywords

• nonn

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.

Examples

			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}.
		

Crossrefs

Cf. A158291, the number of partitions which have hookset {1,2,...,n}, not counting multiplicities.

Programs

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 *)

Extensions

a(31)-a(53) from Alois P. Heinz, Aug 12 2013