A096771 -id:A096771 - cp's OEIS frontend

A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.

1, 0, 1, 0, 2, 0, 3, 0, 4, 1, 0, 5, 2, 0, 6, 5, 0, 7, 8, 0, 8, 14, 0, 9, 20, 1, 0, 10, 30, 2, 0, 11, 40, 5, 0, 12, 55, 10, 0, 13, 70, 18, 0, 14, 91, 30, 0, 15, 112, 49, 0, 16, 140, 74, 1, 0, 17, 168, 110, 2, 0, 18, 204, 158, 5, 0, 19, 240, 221, 10, 0, 20, 285, 302, 20, 0, 21, 330, 407

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.

			Triangle starts:
  1;
  0,  1;
  0,  2;
  0,  3;
  0,  4,  1;
  0,  5,  2;
  0,  6,  5;
  0,  7,  8;
  0,  8, 14;
  0,  9, 20,  1;
  0, 10, 30,  2;
From _Gus Wiseman_, Apr 12 2019: (Start)
Row n = 9 counts the following partitions:
  (9)          (54)       (333)
  (81)         (63)
  (711)        (72)
  (6111)       (432)
  (51111)      (441)
  (411111)     (522)
  (3111111)    (531)
  (21111111)   (621)
  (111111111)  (3222)
               (3321)
               (4221)
               (4311)
               (5211)
               (22221)
               (32211)
               (33111)
               (42111)
               (222111)
               (321111)
               (2211111)
(End)

Alois P. Heinz, Rows n = 0..600, flattened
Findstat, St000183: The side length of the Durfee square of an integer partition
Eric Weisstein's World of Mathematics, Durfee Square

For a version without zeros see A115994. Row lengths are A003059. Row sums are A000041. Column k = 2 is A006918. Column k = 3 is A117485.
Cf. A008284, A115721, A115722, A257990, A325164.
Related triangles are A096771, A325188, A325189, A325192, with Heinz-encoded versions A263297, A325169, A065770, A325178.

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):
seq(seq(T(n, k), k=0..floor(sqrt(n))), n=0..30); # Alois P. Heinz, Apr 09 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := Sum[b[m, k]*b[n-k^2-m, k], {m, 0, n-k^2}]; Table[ T[n, k], {n, 0, 30}, {k, 0, Sqrt[n]}] // Flatten (* Jean-François Alcover, Dec 03 2015, after Alois P. Heinz *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==k&]],{n,0,10},{k,0,Sqrt[n]}] (* Gus Wiseman, Apr 12 2019 *)

T(n,k) = Sum_{i=0..n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.

A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

Original entry on oeis.org

1, 4, 14, 50, 182, 672, 2508, 9438, 35750, 136136, 520676, 1998724, 7696444, 29716000, 115000920, 445962870, 1732525830, 6741529080, 26270128500, 102501265020, 400411345620, 1565841089280, 6129331763880, 24014172955500, 94163002754652, 369507926510352

Offset: 1

Barry E. Williams, Dec 19 1999

Number of partitions with Ferrers plots that fit inside an n X n box, but not in an n-1 X n-1 box. - Wouter Meeussen, Dec 10 2001
From Benoit Cloitre, Jan 29 2002: (Start)
Let m(1,j)=j, m(i,1)=i and m(i,j) = m(i-1,j) + m(i,j-1); then a(n) = m(n,n):
  1  2  3  4 ...
  2  4  7 11 ...
  3  7 14 25 ...
  4 11 25 50 ... (End)
This sequence also gives the number of clusters and non-crossing partitions of type D_n. - F. Chapoton, Jan 31 2005
If Y is a 2-subset of a 2n-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Prefaced with a 1: (1, 1, 4, 14, 50, ...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91, ...). - Gary W. Adamson, May 15 2009
Total number of up steps before the second return in all Dyck n-paths. - David Scambler, Aug 21 2012
Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - Gary Detlefs, Feb 19 2013
First differences of A000984 and A030662. - J. M. Bergot, Jun 22 2013
From R. J. Mathar, Jun 30 2013: (Start)
Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is
  1,   1,   1,   1,   1,   1,
  1,   2,   3,   4,   5,   6,
  1,   3,   6,  10,  15,  21,
  1,   4,  10,  20,  35,  56,
  1,   5,  15,  35,  70, 126,
  1,   6,  21,  56, 126, 252,
and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
From Gus Wiseman, Apr 12 2019: (Start)
Equivalent to Wouter Meeussen's comment, a(n) is the number of integer partitions (of any positive integer) such that the maximum of the length and the largest part is n. For example, the a(1) = 1 through a(3) = 14 partitions are:
  (1)  (2)   (3)
       (11)  (31)
       (21)  (32)
       (22)  (33)
             (111)
             (211)
             (221)
             (222)
             (311)
             (321)
             (322)
             (331)
             (332)
             (333)
(End)
Coxeter-Catalan numbers for Coxeter groups of type D_n [Armstrong]. - N. J. A. Sloane, Mar 09 2022
a(n+1) is the number of ways that a best of n pairs contest with early termination can go. For example, the first stage of an association football (soccer) penalty-kick shoot out has n=5 pairs of shots and there are a(6)=672 distinct ways it can go. For n=2 pairs, writing G for goal and M for miss, and listing the up-to-four shots in chronological order with teams alternating shots, the n(3)=14 possibilities are MMMM, MMMG, MMGM, MMGG, MGM, MGGM, MGGG, GMMM, GMMG, GMG, GGMM, GGMG, GGGM, and GGGG. Not all four shots are taken in two cases because it becomes impossible for one team to overcome the lead of the other team. - Lee A. Newberg, Jul 20 2024

			Sums of {1}, {2, 1, 1}, {2, 2, 3, 3, 2, 1, 1}, {2, 2, 4, 5, 7, 6, 7, 5, 5, 3, 2, 1, 1}, ...

Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 16; See Table 2.8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, arXiv:math/0611106 [math.CO], 2006-2007.
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 10.
F. Chapoton, Clusters.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.
Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions.
Joshua Marsh and Nathan Williams, Nesting Nonpartitions, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, arXiv:1910.14048 [cond-mat.str-el], 2019.
Hugh Thomas, Tamari Lattices and Non-Crossing Partitions in Types B and D, arXiv:math/0311334 [math.CO], 2003-2005.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study, Vol. 56, No. 1 (2023), 1-17.

Left-central elements of the (1, 2)-Pascal triangle A029635.
Column sums of A096771.
Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).
Cf. A128064; first differences of A000984.
Cf. A097613.
Cf. A115720, A252464, A257990, A263297, A325189, A325192, A325193.

a051924 n = a051924_list !! (n-1)
a051924_list = zipWith (-) (tail a000984_list) a000984_list
-- Reinhard Zumkeller, May 25 2013

[Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // Vincenzo Librandi, Dec 21 2016

C:= n-> binomial(2*n, n)/(n+1): seq((n+1)*C(n)-n*C(n-1), n=1..25); # Emeric Deutsch, Jan 08 2008
Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..24); # Zerinvary Lajos, Jan 01 2007
a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):
seq(simplify(a(n)), n=1..24); # Peter Luschny, Dec 14 2015

Table[Binomial[2n,n]-Binomial[2n-2,n-1],{n,30}] (* Harvey P. Dale, Jan 15 2012 *)

a(n)=binomial(2*n,n)-binomial(2*n-2,n-1) \\ Charles R Greathouse IV, Jun 25 2013

{a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1,n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 08 2014

{a(n)=polcoeff( sum(m=1, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(2*m)), n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014

a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))
[a(n) for n in (1..120)] # Peter Luschny, Dec 14 2015

G.f.: (1-x) / sqrt(1-4*x) - 1. - Paul D. Hanna, Nov 08 2014
G.f.: Sum_{n>=1} x^n/(1-x)^(2*n) * Sum_{k=0..n} C(n,k)^2 * x^k. - Paul D. Hanna, Nov 08 2014
a(n+1) = binomial(2*n, n) + 2*Sum_{i=0..n-1} binomial(n+i, i) (V's in Pascal's Triangle). - Jon Perry Apr 13 2004
a(n) = n*C(n-1) - (n-1)*C(n-2), where C(n) = A000108(n) = Catalan(n). For example, a(5) = 50 = 5*C(4) - 4*C(3) - 5*14 - 3*5 = 70 - 20. Triangle A128064 as an infinite lower triangular matrix * A000108 = A051924 prefaced with a 1: (1, 1, 4, 14, 50, 182, ...). - Gary W. Adamson, May 15 2009
Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - Zerinvary Lajos, Dec 20 2005
a(n+1) = A051597(2n,n). - Philippe Deléham, Nov 26 2006
The sequence 1,1,4,... has a(n) = C(2*n,n)-C(2*(n-1),n-1) = 0^n+Sum_{k=0..n} C(n-1,k-1)*A002426(k), and g.f. given by (1-x)/(1-2*x-2*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-.... (continued fraction). - Paul Barry, Oct 17 2009
a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - David Scambler, Aug 21 2012
D-finite with recurrence: a(n) = 2*(3*n-2)*(2*n-3)*a(n-1)/(n*(3*n-5)). - Alois P. Heinz, Apr 25 2014
a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) ~ (3/4)*4^n*(1-(7/24)/n-(7/128)/n^2-(85/3072)/n^3-(581/32768)/n^4-(2611/262144)/n^5)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Dec 20 2016
a(n) = 2 * A097613(n) for n > 1. - Bruce J. Nicholson, Jan 06 2019
Sum_{n>=1} a(n)/8^n = 7/(4*sqrt(2)) - 1. - Amiram Eldar, May 06 2023

Edited by N. J. A. Sloane, May 03 2008, at the suggestion of R. J. Mathar

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

Original entry on oeis.org

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)

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

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.

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A051924 a(n) = binomial(2*n,n) - binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

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

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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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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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A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).