A325200 -id:A325200 - cp's OEIS frontend

A071724 a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.

1, 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, 149226, 534888, 1931540, 7020405, 25662825, 94287120, 347993910, 1289624490, 4796857230, 17902146600, 67016296620, 251577050010, 946844533674, 3572042254128, 13505406670700

Offset: 0

N. J. A. Sloane, Jun 06 2002

Number of standard tableaux of shape (n+1,n-1) (n>=1). - Emeric Deutsch, May 30 2004
From Gus Wiseman, Apr 12 2019: (Start)
Also the number of integer partitions (of any positive integer) such that n is the maximum number of unit steps East or South in the Young diagram starting from the upper-left square and ending in a boundary square in the lower-right quadrant. Also the number of integer partitions fitting in a triangular partition of length n but not of length n - 1. For example, the a(0) = 1 through a(4) = 9 partitions are:
  ()  (1)  (2)   (3)
           (11)  (22)
           (21)  (31)
                 (32)
                 (111)
                 (211)
                 (221)
                 (311)
                 (321)
(End)
The sequence (-1)^(n+1)*a(n), for n >= 1 and +1 for n = 0, is the so-called Z-sequence of the Riordan triangle A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. - Wolfdieter Lang, Oct 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, The a(4)=28 Young tableaux of shape [5, 3].
FindStat, St000384: The maximal part of the shifted composition of an integer partition.
Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 21.
Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
Gus Wiseman, Young diagrams of all integer partitions fitting in a triangular partition of length n but not of length n - 1, n = 1...4.

Cf. A000245, A001622, A002421, A030237, A051924, A096771, A115720, A158909.
Cf. A325169, A325188, A325193, A325195, A325200.
Number of times n appears in A065770.
Column sums of A325189.
Row sums of A030237.

[1] cat [3*Binomial(2*n,n-1)/(n+2): n in [1..29]]; // Vincenzo Librandi, Jul 12 2017

A071724:= n-> 3*binomial(2*n, n-1)/(n+2); 1,seq(A071724(n), n=1..30); # G. C. Greubel, Mar 17 2021

Join[{1}, Table[3Binomial[2n, n-1]/(n+2), {n,1,30}]] (* Vincenzo Librandi, Jul 12 2017 *)
nn=7;
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
allip=Join@@Table[IntegerPartitions[n],{n,0,nn*(nn+1)/2}];
Table[Length[Select[allip,otbmax[#]==n&]],{n,0,nn}] (* Gus Wiseman, Apr 12 2019 *)

a(n)=if(n<1,n==0,3*(2*n)!/(n+2)!/(n-1)!)

[1]+[3*n*catalan_number(n)/(n+2) for n in (1..30)] # G. C. Greubel, Mar 17 2021

a(n) = A000245(n), n>0.
G.f.: (C(x)-1)*(1-x)/x = (1 + x^2 * C(x)^3)*C(x), where C(x) is g.f. for Catalan numbers, A000108.
G.f.: ((1-sqrt(1-4*x))/(2*x)-1)*(1-x)/x = A(x) satisfies x^2*A(x)^2 + (x-1)*(2*x-1)*A(x) + (x-1)^2 = 0.
G.f.: 1 + x*C(x)^3, where C(x) is g.f. for the Catalan numbers (A000108). Sequence without the first term is the 3-fold convolution of the Catalan sequence. - Emeric Deutsch, May 30 2004
a(n) is the n-th moment of the function defined on the segment (0, 4) of x axis: a(n) = Integral_{x=0..4} x^n*(-x^(1/2)*cos(3*arcsin((1/2)*x^(1/2)))/Pi) dx, n=0, 1... . - Karol A. Penson, Sep 29 2004
D-finite with recurrence -(n+2)*(n-1)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jul 10 2017
a(n) ~ c*2^(2*n)*n^(-3/2), where c = 3/sqrt(Pi). - Stefano Spezia, Sep 23 2022
From Amiram Eldar, Sep 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 14*(Pi/(3*sqrt(3)) + 1)/9.
Sum_{n>=0} (-1)^n/a(n) = 18/25 - 164*log(phi)/(75*sqrt(5)), where phi is the golden ratio (A001622). (End)

A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 0, 3, 2, 2, 0, 0, 3, 4, 2, 2, 0, 0, 2, 5, 4, 2, 2, 0, 0, 1, 7, 6, 4, 2, 2, 0, 0, 1, 6, 9, 6, 4, 2, 2, 0, 0, 0, 7, 11, 10, 6, 4, 2, 2, 0, 0, 0, 5, 14, 13, 10, 6, 4, 2, 2, 0, 0, 0, 5, 15, 19, 14, 10, 6, 4, 2, 2, 0, 0, 0, 3, 17, 22, 21, 14, 10, 6, 4, 2, 2, 0, 0, 0, 2, 17, 29

Offset: 1

Wouter Meeussen, Aug 21 2004

Row sums are A000041. Columns are finite and sum to A051924. The final floor(n/2) terms of each row are the reverse of the initial terms of 2*A000041.

			T(5,3)=3, counting 32, 311 and 221.
From _Gus Wiseman_, Apr 12 2019: (Start)
Triangle begins:
  1
  0  2
  0  1  2
  0  1  2  2
  0  0  3  2  2
  0  0  3  4  2  2
  0  0  2  5  4  2  2
  0  0  1  7  6  4  2  2
  0  0  1  6  9  6  4  2  2
  0  0  0  7 11 10  6  4  2  2
  0  0  0  5 14 13 10  6  4  2  2
  0  0  0  5 15 19 14 10  6  4  2  2
  0  0  0  3 17 22 21 14 10  6  4  2  2
  0  0  0  2 17 29 27 22 14 10  6  4  2  2
  0  0  0  1 17 33 36 29 22 14 10  6  4  2  2
  0  0  0  1 15 39 45 41 30 22 14 10  6  4  2  2
  0  0  0  0 14 41 57 52 43 30 22 14 10  6  4  2  2
  0  0  0  0 11 47 67 69 57 44 30 22 14 10  6  4  2  2
  0  0  0  0  9 46 81 85 76 59 44 30 22 14 10  6  4  2  2
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Findstat, FindStat - St000784: The maximum of the length and the largest part of the integer partition

A version with reflected rows is A338621.
Cf. A051924, A096272, A096597, A115994, A252464, A263297, A325193, A368985.
Related triangles are A115720, A325188, A325189, A325192, A325200, with Heinz-encoded versions A257990, A325169, A065770, A325178, A325195.

Table[Count[Partitions[n], q_List /; Max[Length[q], Max[q]]===k], {n, 16}, {k, n}]

row(n)={my(r=vector(n)); forpart(p=n, r[max(#p,p[#p])]++); r} \\ Andrew Howroyd, Jan 12 2024

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

A325188 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0, 0, 2, 5, 0, 0, 0, 0, 2, 8, 1, 0, 0, 0, 0, 2, 9, 4, 0, 0, 0, 0, 0, 2, 12, 8, 0, 0, 0, 0, 0, 0, 2, 13, 15, 0, 0, 0, 0, 0, 0, 0, 2, 16, 23, 1, 0, 0, 0, 0, 0, 0, 0, 2, 17, 32, 5, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Apr 11 2019

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps right 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 the diagram.

			Triangle begins:
  1
  0  1
  0  2  0
  0  2  1  0
  0  2  3  0  0
  0  2  5  0  0  0
  0  2  8  1  0  0  0
  0  2  9  4  0  0  0  0
  0  2 12  8  0  0  0  0  0
  0  2 13 15  0  0  0  0  0  0
  0  2 16 23  1  0  0  0  0  0  0
  0  2 17 32  5  0  0  0  0  0  0  0
  0  2 20 43 12  0  0  0  0  0  0  0  0
  0  2 21 54 24  0  0  0  0  0  0  0  0  0
  0  2 24 67 42  0  0  0  0  0  0  0  0  0  0
  0  2 25 82 66  1  0  0  0  0  0  0  0  0  0  0

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
N. Guru Sharan, Rook decomposition of the Partition function, arXiv:2507.20260 [math.CO], 2025. See p. 4.
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
Eric Weisstein's World of Mathematics, Graph Distance.

Row sums are A000041.
Columns k=1-4 give: A130130, A325168, A382682, A384562.
Cf. A000245, A065770, A096771, A115994, A325169, A325183, A325187, A325189, A325191, A325195, A325200, A368986.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]==k&]],{n,0,15},{k,0,n}]

row(n)={my(r=vector(n+1)); forpart(p=n, my(w=#p); for(i=1, #p, w=min(w,#p-i+p[i])); r[w+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A368986(n).

A325189 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 3, 2, 0, 0, 0, 3, 2, 2, 0, 0, 0, 1, 6, 2, 2, 0, 0, 0, 0, 7, 4, 2, 2, 0, 0, 0, 0, 6, 8, 4, 2, 2, 0, 0, 0, 0, 4, 12, 6, 4, 2, 2, 0, 0, 0, 0, 1, 15, 12, 6, 4, 2, 2, 0, 0, 0, 0, 0, 17, 15, 10, 6, 4, 2, 2

Offset: 0

Gus Wiseman, Apr 11 2019

The maximum origin-to-boundary graph-distance of an integer partition is one plus the maximum number of unit steps East or South in the Young diagram that can be followed, starting from the upper-left square, to reach a boundary square in the lower-right quadrant. It is also the side-length of the minimum triangular partition containing the diagram.

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  0  3  2
  0  0  0  3  2  2
  0  0  0  1  6  2  2
  0  0  0  0  7  4  2  2
  0  0  0  0  6  8  4  2  2
  0  0  0  0  4 12  6  4  2  2
  0  0  0  0  1 15 12  6  4  2  2
  0  0  0  0  0 17 15 10  6  4  2  2
  0  0  0  0  0 14 23 16 10  6  4  2  2
  0  0  0  0  0 10 30 23 14 10  6  4  2  2
  0  0  0  0  0  5 39 29 24 14 10  6  4  2  2
  0  0  0  0  0  1 42 42 31 22 14 10  6  4  2  2
Row 9 counts the following partitions:
  (432)   (54)     (63)      (72)       (81)        (9)
  (3321)  (333)    (621)     (711)      (21111111)  (111111111)
  (4221)  (441)    (6111)    (2211111)
  (4311)  (522)    (222111)  (3111111)
          (531)    (321111)
          (3222)   (411111)
          (5211)
          (22221)
          (32211)
          (33111)
          (42111)
          (51111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Bridget Eileen Tenner, Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), table 1.
Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, Hook length and symplectic content in partitions, arXiv:2205.07322 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Graph Distance

Row sums are A000041. Column sums are A071724.

Cf. A065770, A096771, A115720, A115994, A139582, A325169, A325183, A325188, A325195, A325200, A366157.

otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]==k&]],{n,0,15},{k,0,n}]

row(n)={my(r=vector(n+1)); forpart(p=n, my(w=0); for(i=1, #p, w=max(w,#p-i+p[i])); r[w+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A366157(n). - Andrew Howroyd, Jan 12 2024

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

A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

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

			The partition (3,3) has Heinz number 25 and diagram
  o o o
  o o o
containing maximal triangular partition
  o o
  o
and contained in minimal triangular partition
  o o o o
  o o o
  o o
  o
so a(25) = 4 - 2 = 2.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325183, A325188, A325189, A325191, A325196, A325197, 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]];
Table[otbmax[primeptn[n]]-otb[primeptn[n]],{n,100}]

A325196 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 1.

Original entry on oeis.org

3, 4, 9, 10, 12, 15, 18, 20, 42, 45, 50, 60, 63, 70, 75, 84, 90, 100, 105, 126, 140, 150, 294, 315, 330, 350, 420, 441, 462, 490, 495, 525, 550, 588, 630, 660, 693, 700, 735, 770, 825, 882, 924, 980, 990, 1050, 1100, 1155, 1386, 1470, 1540, 1650, 2730, 3234

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

The enumeration of these partitions by sum is given by A325191.

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   42: {1,2,4}
   45: {2,2,3}
   50: {1,3,3}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
  100: {1,1,3,3}
  105: {2,3,4}
  126: {1,2,2,4}

Gus Wiseman, Young diagrams for their first 60 terms.
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
FindStat, St000384: The maximal part of the shifted composition of an integer partition.

Cf. A060687, A065770, A256617, A325166, A325169, A325179, A325181, A325183, A325185, A325188, A325189, A325191, A325195, 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[#]]==1&]

A325191 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 1.

Original entry on oeis.org

0, 0, 2, 0, 3, 3, 0, 4, 6, 4, 0, 5, 10, 10, 5, 0, 6, 15, 20, 15, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 28, 56, 70, 56, 28, 8, 0, 9, 36, 84, 126, 126, 84, 36, 9, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 11, 55, 165, 330, 462

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325196.

Under the Bulgarian solitaire step, these partitions form cycles of length >= 2. Length >= 2 means not the length=1 self-loop which occurs from the triangular partition when n is a triangular number. See A074909 for self-loops included. - Kevin Ryde, Sep 27 2019

			The a(2) = 2 through a(12) = 10 partitions (empty columns not shown):
  (2)   (22)   (32)   (322)   (332)   (432)   (4322)   (4332)
  (11)  (31)   (221)  (331)   (422)   (3321)  (4331)   (4422)
        (211)  (311)  (421)   (431)   (4221)  (4421)   (4431)
                      (3211)  (3221)  (4311)  (5321)   (5322)
                              (3311)          (43211)  (5331)
                              (4211)                   (5421)
                                                       (43221)
                                                       (43311)
                                                       (44211)
                                                       (53211)

FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition
FindStat, St000384: The maximal part of the shifted composition of an integer partition
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
Eric Weisstein's World of Mathematics, Graph Distance

Column k=1 of A325200.

Cf. A060687, A065770, A071724, A256617, A325166, A325169, A325178, A325179, A325181, A325187, A325188, A325189, A325195, A325196.

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],otb[#]+1==otbmax[#]&]],{n,0,30}]

a(n) = my(t=ceil(sqrtint(8*n+1)/2), r=n-t*(t-1)/2); if(r==0,0, binomial(t,r)); \\ Kevin Ryde, Sep 27 2019

Positions of zeros are A000217 = n * (n + 1) / 2.

a(n) = A074909(n) - A010054(n). - Kevin Ryde, Sep 27 2019

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&]

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}]

cp's OEIS Frontend

A071724 a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.

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A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

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A325188 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k.

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A325189 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.

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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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A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

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A325196 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 1.

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