A307386 -id:A307386 - cp's OEIS frontend

A257990 The side-length of the Durfee square of the partition having Heinz number n.

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1

Offset: 1

Emeric Deutsch, May 18 2015

nonn

The Durfee square of a partition is the largest square that fits inside the Ferrers board of the partition.
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
First appearance of k is a(prime(k)^k) = k. - Gus Wiseman, Apr 12 2019

			a(9)=2; indeed, 9 = 3*3 is the Heinz number of the partition [2,2] and, clearly its Durfee square has side-length =2.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass. 1976.
G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
M. Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Findstat, St000183: The side length of the Durfee square of an integer partition

Positions of 1's are A093641. Positions of 2's are A325164. Positions of 3's are A307386.

Cf. A006918, A056239, A062457, A065770, A112798, A115720, A117485, A215366, A252464, A325163, A325169.

with(numtheory): a := proc (p) local B, S, i: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: S := {}: for i to nops(B(p)) do if i <= B(p)[nops(B(p))+1-i] then S := `union`(S, {i}) else  end if end do: max(S) end proc: seq(a(n), n = 2 .. 146);
# second Maple program:
a:= proc(n) local l, t;
      l:= sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`);
      for t from nops(l) to 1 by -1 do if l[t]>=t then break fi od; t
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, May 10 2016

a[n_] := a[n] = Module[{l, t}, l = Reverse[Sort[Flatten[Table[PrimePi[ f[[1]] ], {f, FactorInteger[n]}, {f[[2]]}]]]]; For[t = Length[l], t >= 1, t--, If[l[[t]] >= t, Break[]]]; t]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Feb 17 2017, after Alois P. Heinz *)

For a partition (p_1 >= p_2 >= ... > = p_r) the side-length of its Durfee square is the largest i such that p_i >=i.

a(1)=0 prepended by Alois P. Heinz, May 10 2016

A117485 Expansion of x^9/((1-x)(1-x^2)(1-x^3))^2.

Original entry on oeis.org

1, 2, 5, 10, 18, 30, 49, 74, 110, 158, 221, 302, 407, 536, 698, 896, 1136, 1424, 1770, 2176, 2656, 3216, 3866, 4616, 5481, 6466, 7591, 8866, 10306, 11926, 13747, 15778, 18046, 20566, 23359, 26446, 29855, 33600, 37716, 42224, 47152, 52528, 58388, 64752, 71664

Offset: 9

Alford Arnold, Mar 22 2006

Molien series for S_3 X S_3, cf. A001399.
From Gus Wiseman, Apr 06 2019: (Start)
Also the number of integer partitions of n with Durfee square of length 3. The Heinz numbers of these partitions are given by A307386. For example, the a(9) = 1 through a(13) = 18 partitions are:
  (333)  (433)   (443)    (444)     (544)
         (3331)  (533)    (543)     (553)
                 (3332)   (633)     (643)
                 (4331)   (3333)    (733)
                 (33311)  (4332)    (4333)
                          (4431)    (4432)
                          (5331)    (4441)
                          (33321)   (5332)
                          (43311)   (5431)
                          (333111)  (6331)
                                    (33322)
                                    (33331)
                                    (43321)
                                    (44311)
                                    (53311)
                                    (333211)
                                    (433111)
                                    (3331111)
(End)

			As a cross-check, row sixteen of A115994 yields p(16) = 16 + 140 + 74 + 1.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).
Index entries for two-way infinite sequences

Column k=3 of A115994.
Cf. A000027 (for k=1), A006918 (for k=2), A117488, A117489, A001399, A117486.
Cf. A115720, A307386, A325164.

n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane, Mar 10 2007

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=3, stack): seq(count(subs(r=3, ZL), size=m), m=6..50) ; # Zerinvary Lajos, Jan 02 2008

CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3))^2,{x,0,50}],x] (* Harvey P. Dale, Oct 09 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==3&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Dec 12 2019

a(n) = floor((3*n^5 - 45*n^4 + 200*n^3 - 180*n^2 - 363*n + 1600)/12960 + n/27*(n%3==0) - n/32*(n%2==0)) \\ Hoang Xuan Thanh, Jul 17 2025

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>20. - Colin Barker, Dec 12 2019
From Hoang Xuan Thanh, May 17 2025: (Start)
a(n+3) = Sum_{x+2*y+3*z=n} x*y*z.
a(n+3) = n*(n^2-1)*(3*n^2-67)/12960 - floor((n+1)/3)/27 + [n mod 2 = 0]*n/32 + [n mod 3 = 0]*n/27 where [] is the Iverson bracket. (End)

Entry revised by N. J. A. Sloane, Mar 10 2007

A307515 Heinz numbers of integer partitions with Durfee square of length > 2.

Original entry on oeis.org

125, 175, 245, 250, 275, 325, 343, 350, 375, 385, 425, 455, 475, 490, 500, 525, 539, 550, 575, 595, 605, 625, 637, 650, 665, 686, 700, 715, 725, 735, 750, 770, 775, 805, 825, 833, 845, 847, 850, 875, 910, 925, 931, 935, 950, 975, 980, 1000, 1001, 1015, 1025

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

First differs from A307386 in having 7^4 = 2401.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The Durfee square of an integer partition is the largest square contained in its Young diagram.
The enumeration of these partitions by sum is given by A084835.

			The sequence of terms together with their prime indices begins:
  125: {3,3,3}
  175: {3,3,4}
  245: {3,4,4}
  250: {1,3,3,3}
  275: {3,3,5}
  325: {3,3,6}
  343: {4,4,4}
  350: {1,3,3,4}
  375: {2,3,3,3}
  385: {3,4,5}
  425: {3,3,7}
  455: {3,4,6}
  475: {3,3,8}
  490: {1,3,4,4}
  500: {1,1,3,3,3}
  525: {2,3,3,4}
  539: {4,4,5}
  550: {1,3,3,5}
  575: {3,3,9}
  595: {3,4,7}

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Gus Wiseman, Table of n, a(n) for n = 1..22485
FindStat, St000183: The side length of the Durfee square of an integer partition
Wikipedia, Durfee square.

Positions of numbers > 2 in A257990.

Cf. A006918, A056239, A084835, A112798, A115994, A117485, A252464, A325163, A325170.

durf[n_]:=Length[Select[Range[PrimeOmega[n]], Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]][[#]]>=#&]];
Select[Range[100], durf[#]>2&]

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A257990 The side-length of the Durfee square of the partition having Heinz number n.

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A117485 Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2.

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A307515 Heinz numbers of integer partitions with Durfee square of length > 2.

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A117485 Expansion of x^9/((1-x)(1-x^2)(1-x^3))^2.