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

0,3

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

Table of n, a(n) for n=0..53.

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: A029040 A053281 A339395 * 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