A325188 -id:A325188 - cp's OEIS frontend

A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

0, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0, 0, 2, 4, 1, 0, 0, 0, 2, 6, 3, 0, 0, 0, 0, 2, 6, 6, 1, 0, 0, 0, 0, 2, 8, 9, 3, 0, 0, 0, 0, 0, 2, 8, 13, 6, 1, 0, 0, 0, 0, 0, 2, 10, 16, 11, 3, 0, 0, 0, 0, 0, 0, 2, 10, 20, 17, 6, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 12 2019

			Triangle begins:
  1
  2  0
  2  1  0
  2  3  0  0
  2  4  1  0  0
  2  6  3  0  0  0
  2  6  6  1  0  0  0
  2  8  9  3  0  0  0  0
  2  8 13  6  1  0  0  0  0
  2 10 16 11  3  0  0  0  0  0
  2 10 20 17  6  1  0  0  0  0  0
  2 12 24 25 11  3  0  0  0  0  0  0
  2 12 28 33 19  6  1  0  0  0  0  0  0
  2 14 32 44 29 11  3  0  0  0  0  0  0  0
  2 14 38 53 43 19  6  1  0  0  0  0  0  0  0
Row n = 9 counts the following partitions:
  (9)          (54)        (333)      (4221)    (51111)
  (111111111)  (63)        (432)      (4311)
               (72)        (441)      (5211)
               (81)        (522)      (6111)
               (22221)     (531)      (42111)
               (222111)    (621)      (411111)
               (2211111)   (711)
               (21111111)  (3222)
                           (3321)
                           (32211)
                           (33111)
                           (321111)
                           (3111111)

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Column k = 2 is A265283. Column k = 3 is A325228.

Cf. A051924, A096771, A115720, A263297, A325188, A325189, A325192, A325193, A325194, A325224, A325225, A325229, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==k&]],{n,15},{k,n}]

A325199 Number of integer partitions of n 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

0, 0, 0, 2, 0, 2, 6, 3, 2, 9, 15, 12, 6, 12, 27, 38, 34, 22, 20, 43, 74, 94, 90, 67, 48, 69, 130, 194, 232, 230, 187, 132, 129, 218, 364, 497, 576, 578, 498, 367, 290, 378, 642, 977, 1264, 1435, 1448, 1290, 1000, 735, 728

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325197.

			The a(3) = 2 through a(10) = 15 partitions (empty columns not shown):
  (3)    (41)    (33)    (43)    (521)    (333)    (433)
  (111)  (2111)  (42)    (2221)  (32111)  (441)    (442)
                 (222)   (4111)           (522)    (532)
                 (411)                    (531)    (541)
                 (2211)                   (3222)   (3322)
                 (3111)                   (5211)   (3331)
                                          (32211)  (4222)
                                          (33111)  (4411)
                                          (42111)  (5221)
                                                   (5311)
                                                   (32221)
                                                   (33211)
                                                   (42211)
                                                   (43111)
                                                   (52111)

Column k=2 of A325200.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325188, A325189, A325191, A325195, A325197, A325198.

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

A382682 Number of integer partitions of n with origin-to-boundary graph-distance equal to 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 8, 15, 23, 32, 43, 54, 67, 82, 97, 114, 133, 152, 173, 196, 219, 244, 271, 298, 327, 358, 389, 422, 457, 492, 529, 568, 607, 648, 691, 734, 779, 826, 873, 922, 973, 1024, 1077, 1132, 1187, 1244, 1303, 1362, 1423, 1486, 1549, 1614, 1681, 1748, 1817, 1888, 1959

Offset: 0

N Guru Sharan, Jun 03 2025

Also the number of partitions of n with a fixed Durfee triangle of size 3.

Column k=3 of the triangle in A325188.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
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 (2,-1,1,-2,1).

Cf. A130130, A325168, A325188.

CoefficientList[Series[(q^6 + 2 q^7 + q^8 + 2 q^9 - q^10 - q^12 - q^13 + q^14)/((1 - q)^3 (1 + q + q^2)), {q, 0, 200}],q]

G.f.: q^6*(1 + 2*q + q^2 + 2*q^3 - q^4 - q^6 - q^7 + q^8)/((1 - q)^3*(1 + q + q^2)).

9*a(n) = 2*A099837(n+3)+6*n^2+59-45*n for n>9. - R. J. Mathar, Jun 24 2025

A368986 a(n) = sum of the origin-to-boundary graph-distances of all partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 8, 12, 21, 32, 50, 73, 107, 152, 219, 302, 419, 567, 771, 1027, 1374, 1806, 2375, 3083, 3999, 5136, 6597, 8398, 10676, 13477, 16981, 21260, 26584, 33057, 41049, 50738, 62605, 76930, 94374, 115330, 140704, 171106, 207732, 251460, 303919, 366335, 440880, 529298

Offset: 0

Andrew Howroyd, Jan 12 2024

nonn

The origin-to-boundary graph-distance (see A325188) is the side length of the maximum triangular partition contained inside the Ferrer's diagram of the partition. a(n) is the sum of the side lengths over all partitions of n.

Cf. A325188, A325189, A325200.

a(n)={my(s=0); forpart(p=n, my(w=#p); for(i=1, #p, w=min(w, #p-i+p[i])); s += w); s}

a(n) = Sum_{k=1..n} k*A325188(n,k).

A384562 Number of integer partitions of n with origin-to-boundary graph-distance equal to 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 12, 24, 42, 66, 98, 135, 181, 233, 298, 367, 452, 543, 651, 765, 899, 1039, 1202, 1371, 1564, 1765, 1993, 2227, 2491, 2763, 3066, 3377, 3722, 4075, 4465, 4863, 5299, 5745, 6232, 6727, 7266, 7815, 8409, 9013, 9665, 10327, 11040, 11763, 12538, 13325, 14167, 15019, 15929, 16851, 17832, 18825, 19880, 20947, 22079, 23223, 24433, 25657, 26950

Offset: 0

N Guru Sharan, Jun 03 2025

This also counts the number of partitions of n with a fixed Durfee triangle of size 4. This is the column k=4 of the triangle in A325188.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A130130, A325168, A325188, A382682.

CoefficientList[Series[(q^10 (1 + 4q + 6 q^2 + 7 q^3 + 6 q^4 + 2 q^5 - 5 q^7 - 5 q^8 - 5 q^9 + q^11 + 3 q^12 + 2 q^13 - q^16))/((1 - q)(1 - q^2)(1 - q^3)(1 - q^4)), {q, 0, 50}], q]

G.f.: q^10*(1 + 4*q + 6*q^2 + 7*q^3 + 6*q^4 + 2*q^5 - 5*q^7 - 5*q^8 - 5*q^9 + q^11 + 3*q^12 + 2*q^13 - q^16)/((1 - q)*(1 - q^2)*(1 - q^3)*(1 - q^4)).

cp's OEIS Frontend

A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A325199 Number of integer partitions of n 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A382682 Number of integer partitions of n with origin-to-boundary graph-distance equal to 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A368986 a(n) = sum of the origin-to-boundary graph-distances of all partitions of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A384562 Number of integer partitions of n with origin-to-boundary graph-distance equal to 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula