A325180 -id:A325180 - cp's OEIS frontend

A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

0, 0, 1, 1, 2, 1, 3, 2, 0, 2, 4, 2, 5, 3, 1, 3, 6, 1, 7, 2, 2, 4, 8, 3, 1, 5, 1, 3, 9, 1, 10, 4, 3, 6, 2, 2, 11, 7, 4, 3, 12, 2, 13, 4, 1, 8, 14, 4, 2, 1, 5, 5, 15, 2, 3, 3, 6, 9, 16, 2, 17, 10, 2, 5, 4, 3, 18, 6, 7, 2, 19, 3, 20, 11, 1, 7, 3, 4, 21, 4, 2, 12

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition (3,3,2,1) has Heinz number 150 and diagram
  o o o
  o o o
  o o
  o
containing maximal square
  o o
  o o
and contained in minimal square
  o o o o
  o o o o
  o o o o
  o o o o
so a(150) = 4 - 2 = 2.

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

Wikipedia, Durfee square.

Positions of zeros are A062457. Positions of 1's are A325179. Positions of 2's are A325180.

Cf. A001222, A046660, A051924, A056239, A061395, A093641, A096771, A115994, A243055, A257990, A263297, A325192, A325195.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Table[codurf[n]-durf[n],{n,100}]

a(n) = A263297(n) - A257990(n).

A325192 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 2, 2, 0, 0, 2, 1, 2, 2, 0, 0, 3, 2, 2, 2, 2, 0, 0, 2, 4, 3, 2, 2, 2, 0, 0, 1, 7, 4, 4, 2, 2, 2, 0, 1, 0, 6, 8, 5, 4, 2, 2, 2, 0, 0, 2, 5, 11, 8, 6, 4, 2, 2, 2, 0, 0, 3, 4, 12, 12, 9, 6, 4, 2, 2, 2, 0, 0, 4, 5, 13, 17, 12, 10, 6, 4, 2, 2, 2, 0

Offset: 0

Gus Wiseman, Apr 08 2019

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

			Triangle begins:
  1
  1  0
  0  2  0
  0  1  2  0
  1  0  2  2  0
  0  2  1  2  2  0
  0  3  2  2  2  2  0
  0  2  4  3  2  2  2  0
  0  1  7  4  4  2  2  2  0
  1  0  6  8  5  4  2  2  2  0
  0  2  5 11  8  6  4  2  2  2  0
  0  3  4 12 12  9  6  4  2  2  2  0
  0  4  5 13 17 12 10  6  4  2  2  2  0
  0  3  9 12 20 18 13 10  6  4  2  2  2  0
  0  2 12 15 23 25 18 14 10  6  4  2  2  2  0
  0  1 15 19 26 30 26 19 14 10  6  4  2  2  2  0
Row 9 counts the following partitions (empty columns not shown):
   333   432    54      63       72        711       81         9
         441    522     621      6111      3111111   21111111   111111111
         3222   531     51111    411111
         3321   5211    222111   2211111
         4221   22221   321111
         4311   32211
                33111
                42111

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

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Wikipedia, Durfee square.

Row sums are A000041. Column k = 1 is A325181. Column k = 2 is A325182.
Cf. A096771, A257990, A263297, A325178, A325179, A325180, A325200.
Cf. A115995, A368985.

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==k&]],{n,0,15},{k,0,n}]

row(n)={my(r=vector(n+1)); if(n==0, r[1]=1, forpart(p=n, my(c=1); while(c<#p && cAndrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A368985(n) - A115995(n). - Andrew Howroyd, Jan 12 2024

A325197 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

Original entry on oeis.org

5, 8, 14, 21, 24, 25, 27, 28, 35, 36, 40, 54, 56, 66, 98, 99, 110, 120, 125, 132, 135, 147, 154, 165, 168, 175, 180, 189, 196, 198, 200, 220, 225, 231, 245, 250, 252, 264, 270, 275, 280, 297, 300, 308, 375, 378, 385, 390, 392, 396, 440, 450, 500, 546, 585, 594

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

The enumeration of these partitions by sum is given by A325199.

			The sequence of terms together with their prime indices begins:
    5: {3}
    8: {1,1,1}
   14: {1,4}
   21: {2,4}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   35: {3,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   66: {1,2,5}
   98: {1,4,4}
   99: {2,2,5}
  110: {1,3,5}
  120: {1,1,1,2,3}
  125: {3,3,3}
  132: {1,1,2,5}

Cf. A195086, A065770, A325166, A325168, A325169, A325170, A325180, A325182, A325188, A325189, A325195, A325196, A325198, A325199, A325200.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[1000],otbmax[primeptn[#]]-otb[primeptn[#]]==2&]

A325179 Heinz numbers of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

Original entry on oeis.org

3, 4, 6, 15, 18, 25, 27, 30, 45, 50, 75, 175, 245, 250, 343, 350, 375, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 3773, 4802, 5929, 7203, 7546, 9317, 11319, 11858, 12005, 14641, 16807, 17787, 18634, 18865, 26411, 27951, 29282, 29645, 41503, 43923, 46585

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The enumeration of these partitions by sum is given by A325181.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
    6: {1,2}
   15: {2,3}
   18: {1,2,2}
   25: {3,3}
   27: {2,2,2}
   30: {1,2,3}
   45: {2,2,3}
   50: {1,3,3}
   75: {2,3,3}
  175: {3,3,4}
  245: {3,4,4}
  250: {1,3,3,3}
  343: {4,4,4}
  350: {1,3,3,4}
  375: {2,3,3,3}
  490: {1,3,4,4}
  525: {2,3,3,4}
  625: {3,3,3,3}

Gus Wiseman, Young diagrams for the first 32 terms.

Numbers k such that A263297(k) - A257990(k) = 1.
Positions of 1's in A325178.
Cf. A056239, A093641, A112798, A325180, A325181, A325192, A325196, A325198.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Select[Range[1000],codurf[#]-durf[#]==1&]

A325182 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

Original entry on oeis.org

0, 0, 0, 2, 2, 1, 2, 4, 7, 6, 5, 4, 5, 9, 12, 15, 14, 12, 10, 9, 11, 15, 21, 24, 28, 26, 24, 20, 18, 17, 19, 25, 31, 38, 42, 46, 44, 41, 36, 32, 29, 28, 31, 37, 46, 53, 62, 66, 71, 68, 65, 58, 53, 47, 44, 43, 46, 54, 63, 74, 83, 93, 98, 103, 100, 96, 88, 81

Offset: 0

Gus Wiseman, Apr 08 2019

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

			The a(3) = 2 through a(14) = 12 partitions:
  3    31   311  42    43    44    432   442   533    543    544    554
  111  211       2211  421   422   441   3322  4322   4422   553    5333
                       2221  431   3222  4222  4421   5331   5332   5432
                       3211  2222  3321  4321  33311  33321  5431   5441
                             3221  4221  4411         43311  33322  5531
                             3311  4311                      33331  33332
                             4211                            43321  43322
                                                             44311  43331
                                                             53311  44321
                                                                    44411
                                                                    53321
                                                                    54311

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

Wikipedia, Durfee square.

Cf. A006918, A096771, A257990, A263297, A325168.

Cf. A325178, A325180, A325181, A325182, A325192, A325197, A325199, A325200.

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==2&]],{n,0,30}]

A325198 Positive numbers whose maximum prime index minus minimum prime index is 2.

Original entry on oeis.org

10, 20, 21, 30, 40, 50, 55, 60, 63, 80, 90, 91, 100, 105, 120, 147, 150, 160, 180, 187, 189, 200, 240, 247, 250, 270, 275, 300, 315, 320, 360, 385, 391, 400, 441, 450, 480, 500, 525, 540, 551, 567, 600, 605, 637, 640, 713, 720, 735, 750, 800, 810, 900, 945

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

Also Heinz numbers of integer partitions whose maximum minus minimum part is 2 (counted by A008805). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   20: {1,1,3}
   21: {2,4}
   30: {1,2,3}
   40: {1,1,1,3}
   50: {1,3,3}
   55: {3,5}
   60: {1,1,2,3}
   63: {2,2,4}
   80: {1,1,1,1,3}
   90: {1,2,2,3}
   91: {4,6}
  100: {1,1,3,3}
  105: {2,3,4}
  120: {1,1,1,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  160: {1,1,1,1,1,3}
  180: {1,1,2,2,3}
  187: {5,7}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 2's in A243055.
A061395(a(n)) - A055396((n)) = 2.
Cf. A000961, A008805, A046660, A056239, A093641, A112798, A118914, A174090, A195086, A256617, A325180, A325197.

N:= 1000: # for terms <= N
q:= 2: r:= 3:
Res:= NULL:
do
  p:= q; q:= r; r:= nextprime(r);
  if p*r > N then break fi;
  for i from 1 do
    pi:= p^i;
    if pi*r > N then break fi;
    for j from 0 do
      piqj:= pi*q^j;
      if piqj*r > N then break fi;
      Res:= Res, seq(piqj*r^k,k=1 .. floor(log[r](N/piqj)))
    od
  od
od:
sort([Res]); # Robert Israel, Apr 12 2019

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]==2&]

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A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

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A325192 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k.

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A325197 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

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A325179 Heinz numbers of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

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A325182 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

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A325198 Positive numbers whose maximum prime index minus minimum prime index is 2.

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