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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

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-Fran├žois 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.)