A325170 -id:A325170 - cp's OEIS frontend

A325169 Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the minimum triangular partition contained inside the diagram.

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

FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Cf. A001221, A001222, A056239, A061395, A065770, A112798, A115994, A174090, A257990, A297113, A325166, A325167, A325170.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[otb[Reverse[primeMS[n]]],{n,100}]

A257990(n) <= a(n) <= 2 * A257990(n).

A325164 Heinz numbers of integer partitions with Durfee square of length 2.

Original entry on oeis.org

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 126, 129, 130, 132, 133, 135, 138, 140

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also positions of 2 in A257990.
First differs from A105441 in lacking 125.
The Durfee length 1 case is A093641. The enumeration of Durfee length 2 partitions by sum is given by A006918, while that of Durfee length 3 partitions is given by A117485.

			The sequence of terms together with their prime indices begins:
   9: {2,2}
  15: {2,3}
  18: {1,2,2}
  21: {2,4}
  25: {3,3}
  27: {2,2,2}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  39: {2,6}
  42: {1,2,4}
  45: {2,2,3}
  49: {4,4}
  50: {1,3,3}
  51: {2,7}
  54: {1,2,2,2}
  55: {3,5}
  57: {2,8}
  60: {1,1,2,3}

Gus Wiseman, Young diagrams corresponding to the first 36 terms.

Cf. A006918, A056239, A093641, A112798, A115994, A117485, A252464, A257990, 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&]

A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121

Offset: 0

Gus Wiseman, Apr 05 2019

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside it.

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)
        (31)   (41)    (42)     (52)      (53)       (63)        (64)
        (211)  (221)   (51)     (61)      (62)       (72)        (73)
               (311)   (222)    (511)     (71)       (81)        (82)
               (2111)  (411)    (2221)    (611)      (711)       (91)
                       (2211)   (4111)    (2222)     (6111)      (811)
                       (3111)   (22111)   (5111)     (22221)     (7111)
                       (21111)  (31111)   (22211)    (51111)     (22222)
                                (211111)  (41111)    (222111)    (61111)
                                          (221111)   (411111)    (222211)
                                          (311111)   (2211111)   (511111)
                                          (2111111)  (3111111)   (2221111)
                                                     (21111111)  (4111111)
                                                                 (22111111)
                                                                 (31111111)
                                                                 (211111111)

Colin Barker, Table of n, a(n) for n = 0..1000
N. Guru Sharan, Rook decomposition of the Partition function, arXiv:2507.20260 [math.CO], 2025. See p. 5.
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006918, A065770, A115994, A117485, A257990, A297113, A325135, A325166, A325169, A325170.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]==2&]],{n,0,30}]

concat([0,0,0], Vec(x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)) + O(x^80))) \\ Colin Barker, Apr 08 2019

From Colin Barker, Apr 08 2019: (Start)
G.f.: x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7.
a(n) = 2*n - 4 for n>4 and even.
a(n) = 2*n - 5 for n>4 and odd.
(End)

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&]

A325180 Heinz number 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 2.

Original entry on oeis.org

5, 8, 10, 12, 20, 21, 35, 36, 42, 49, 54, 60, 63, 70, 81, 84, 90, 98, 100, 105, 126, 135, 140, 147, 150, 189, 196, 210, 225, 275, 294, 315, 385, 441, 500, 539, 550, 605, 700, 750, 770, 825, 847, 980, 1050, 1078, 1100, 1125, 1155, 1210, 1250, 1331, 1372, 1375

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The enumeration of these partitions by sum is given by A325182.

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:
    5: {3}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   20: {1,1,3}
   21: {2,4}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   49: {4,4}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   81: {2,2,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  105: {2,3,4}

Gus Wiseman, Young diagrams corresponding to the first 96 terms.

Numbers k such that A263297(k) - A257990(k) = 2.
Positions of 2's in A325178.
Cf. A006918, A056239, A093641, A112798, A325164, A325170, A325179, A325182, A325192, A325197.

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[#]==2&]

A307386 Heinz numbers of integer partitions with Durfee square of length 3.

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 06 2019

nonn

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 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}

Gus Wiseman, Young diagrams corresponding to the first 36 terms.

Positions of 3 in A257990. The Durfee length 1 case is A093641. The Durfee length 2 case is A325164. The enumeration of Durfee length 2 partitions by sum is given by A006918, while that of Durfee length 3 partitions is given by A117485.

Cf. A056239, A112798, A115994, 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[#]==3&]

A325186 Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.

Original entry on oeis.org

3, 4, 15, 18, 21, 25, 27, 33, 36, 39, 51, 57, 69, 72, 87, 93, 105, 111, 123, 129, 141, 144, 147, 150, 159, 165, 175, 177, 183, 195, 201, 213, 219, 225, 231, 237, 245, 249, 250, 255, 267, 273, 275, 285, 288, 291, 300, 303, 309, 321, 325, 327, 339, 343, 345, 357

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary.

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}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   51: {2,7}
   57: {2,8}
   69: {2,9}
   72: {1,1,1,2,2}
   87: {2,10}
   93: {2,11}
  105: {2,3,4}
  111: {2,12}
  123: {2,13}
  129: {2,14}

Eric Weisstein's World of Mathematics, Graph Distance.
Gus Wiseman, Young diagrams for the first 25 terms.

Cf. A006918, A056239, A065770, A112798.

Cf. A325164, A325169, A325170, A325183, A325184, A325185, A325190, A325197.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];
corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];
Select[Range[100],Apply[Plus,If[#==1,{},FixedPointList[corpos,ptnmat[primeptn[#]]][[-3]]],{0,1}]==2&]

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&]

A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

Original entry on oeis.org

27, 45, 54, 63, 75, 81, 90, 99, 105, 108, 117, 126, 135, 147, 150, 153, 162, 165, 171, 180, 189, 195, 198, 207, 210, 216, 225, 231, 234, 243, 252, 255, 261, 270, 273, 279, 285, 294, 297, 300, 306, 315, 324, 330, 333, 342, 345, 351, 357, 360, 363, 369, 378, 387

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

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

The enumeration of these partitions by sum is given by A006918 (see Emeric Deutsch's comment there).

			The sequence of terms together with their prime indices begins:
   27: {2,2,2}
   45: {2,2,3}
   54: {1,2,2,2}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   90: {1,2,2,3}
   99: {2,2,5}
  105: {2,3,4}
  108: {1,1,2,2,2}
  117: {2,2,6}
  126: {1,2,2,4}
  135: {2,2,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  153: {2,2,7}
  162: {1,2,2,2,2}
  165: {2,3,5}
  171: {2,2,8}
  180: {1,1,2,2,3}

Cf. A000726, A002620, A004250, A006918, A056239, A097701, A112798, A257990, A297113, A325164, A325169, A325170.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]>=3&&Reverse[primeMS[#]][[3]]==2&]

cp's OEIS Frontend

A325169 Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.

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A325164 Heinz numbers of integer partitions with Durfee square of length 2.

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A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

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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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A325180 Heinz number 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 2.

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A307386 Heinz numbers of integer partitions with Durfee square of length 3.

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A325186 Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.

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

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