A325184 -id:A325184 - cp's OEIS frontend

A325187 Number of integer partitions of n such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.

1, 0, 1, 3, 3, 5, 9, 14, 20, 26, 38, 53, 75, 101, 132, 175, 229, 301, 394, 509, 650, 826, 1043, 1315, 1656, 2074, 2590, 3218, 3975, 4896, 6008, 7361, 8989, 10960, 13323, 16159, 19531, 23553, 28323, 34002, 40723, 48694, 58115, 69249, 82350, 97766, 115832

Offset: 1

Gus Wiseman, Apr 11 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 sequence gives the number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 1.

The Heinz numbers of these partitions are given by A325185.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (21)  (22)   (41)    (51)     (61)      (71)
             (31)   (311)   (321)    (322)     (332)
             (211)  (2111)  (411)    (331)     (422)
                            (3111)   (421)     (431)
                            (21111)  (511)     (521)
                                     (3211)    (611)
                                     (4111)    (3221)
                                     (31111)   (3311)
                                     (211111)  (4211)
                                               (5111)
                                               (32111)
                                               (41111)
                                               (311111)
                                               (2111111)

Eric Weisstein's World of Mathematics, Graph Distance.

Cf. A000245, A188674, A325165, A325169, A325183, A325184, A325185, A325187, A325190, A325191.

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

A325183 Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 10, 10, 11, 10, 13, 14, 15, 7, 17, 15, 19, 14, 21, 22, 23, 14, 21, 26, 21, 22, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 22, 41, 42, 43, 26, 42, 46, 47, 22, 55, 42, 51, 34, 53, 35, 55, 26, 57, 58, 59, 42, 61, 62, 66, 13, 65, 66, 67

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 partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with origin-to-boundary graph-distances
  4 4 4 3 2 1
  3 3 3 2 1
  2 2 2 1 1
  1 1 1
giving the origin-to-boundary partition (7,5,4,3) with Heinz number 6545, so a(7865) = 6545.

Eric Weisstein's World of Mathematics, Graph Distance.

The only terms appearing only once are the primorials A002110.
The union consists of all squarefree numbers A005117.
Cf. A000245, A056239, A065770, A112798, A174090, A297113.
Cf. A325166, A325167, A325169, A325184, A325188, A325189, A325195.

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];
Table[Times@@Prime/@If[n==1,{},-Differences[Map[Total,Drop[FixedPointList[corpos,ptnmat[primeptn[n]]],-1],2]]],{n,30}]

A325185 Heinz numbers of integer partitions such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.

Original entry on oeis.org

2, 6, 9, 10, 12, 14, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 45, 46, 48, 50, 52, 56, 58, 60, 62, 63, 66, 68, 70, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 110, 112, 114, 116, 117, 118, 120, 122, 124, 125, 126, 130, 132

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 sequence gives all Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 1.

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:
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
   48: {1,1,1,1,2}

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

Cf. A001222, A056239, A061395, A065770, A112798, A188674.

Cf. A325169, A325183, A325184, A325186, A325187, A325196.

hptn[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]];
Select[Range[2,100],otb[hptn[#]]>otb[Rest[hptn[#]]]&&otb[hptn[#]]>otb[DeleteCases[hptn[#]-1,0]]&]

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&]

A325190 Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 4, 2, 2, 4, 8, 10, 12, 10, 14, 20, 28, 36, 44, 46, 56, 66, 86, 108, 136, 160, 190, 214, 252, 298, 364, 434, 524, 620, 728, 834, 966, 1112, 1306, 1522, 1788, 2088, 2448, 2822, 3256, 3720, 4264, 4876, 5610, 6434, 7420

Offset: 0

Gus Wiseman, Apr 11 2019

nonn

The Heinz numbers of these partitions are given by A325186.
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. For example, the partition (6,5,5,3) has diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with origin-to-boundary graph-distances
  4 4 4 3 2 1
  3 3 3 2 1
  2 2 2 1 1
  1 1 1
giving the origin-to-boundary partition (7,5,4,3).

			The a(2) = 1 through a(11) = 10 partitions:
  (2)   (32)   (33)    (52)     (62)      (72)       (82)        (92)
  (11)  (221)  (42)    (22111)  (221111)  (432)      (433)       (443)
               (222)                      (3321)     (442)       (533)
               (2211)                     (2211111)  (532)       (542)
                                                     (3322)      (632)
                                                     (3331)      (3332)
                                                     (33211)     (33221)
                                                     (22111111)  (33311)
                                                                 (332111)
                                                                 (221111111)

Eric Weisstein's World of Mathematics, Graph Distance.

Cf. A188674, A325165, A325169, A325183, A325184, A325186, A325187, A325190, A325199.

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];
Table[Length[Select[IntegerPartitions[n],Apply[Plus,If[#=={},{},FixedPointList[corpos,ptnmat[#]][[-3]]],{0,1}]==2&]],{n,30}]

cp's OEIS Frontend

A325187 Number of integer partitions of n such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A325183 Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A325185 Heinz numbers of integer partitions such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A325190 Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica