A115720 -id:A115720 - cp's OEIS frontend

A188674 Stack polyominoes with square core.

1, 1, 0, 0, 1, 2, 3, 4, 5, 7, 9, 13, 17, 24, 31, 42, 54, 71, 90, 117, 147, 188, 236, 298, 371, 466, 576, 716, 882, 1088, 1331, 1633, 1987, 2422, 2935, 3557, 4290, 5177, 6216, 7465, 8932, 10682, 12731, 15169, 18016, 21387, 25321, 29955, 35353, 41696, 49063, 57689, 67698, 79375, 92896, 108633, 126817, 147922, 172272

Offset: 0

Emanuele Munarini, Apr 08 2011

nonn

a(n) is the number of stack polyominoes of area n with square core.
The core of stack is the set of all maximal columns.
The core is a square when the number of columns is equal to their height.
Equivalently, a(n) is the number of unimodal compositions of n, where the number of the parts of maximum value equal the maximum value itself. For instance, for n = 10, we have the following stacks:
(1,3,3,3), (3,3,3,1), (1,1,1,1,1,1,2,2), (1,1,1,1,1,2,2,1), (1,1,1,1,2,2,1,1), (1,1,1,2,2,1,1,1), (1,1,2,2,1,1,1,1), (1,2,2,1,1,1,1,1), (2,2,1,1,1,1,1,1).
From Gus Wiseman, Apr 06 2019 and May 21 2022: (Start)
Also the number of integer partitions of n with final part in their inner lining partition equal to 1, where the k-th part of the inner lining partition of a partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. For example, the a(4) = 1 through a(10) = 9 partitions are:
  (22)  (32)   (42)    (52)     (62)      (72)       (82)
        (221)  (321)   (421)    (521)     (333)      (433)
               (2211)  (3211)   (4211)    (621)      (721)
                       (22111)  (32111)   (5211)     (3331)
                                (221111)  (42111)    (6211)
                                          (321111)   (52111)
                                          (2211111)  (421111)
                                                     (3211111)
                                                     (22111111)
Also partitions that have a fixed point and a conjugate fixed point, ranked by A353317. The strict case is A352829. For example, the a(0) = 0 through a(9) = 7 partitions are:
  ()  .  .  (21)  (31)   (41)    (51)     (61)      (71)
                  (211)  (311)   (411)    (511)     (332)
                         (2111)  (3111)   (4111)    (611)
                                 (21111)  (31111)   (5111)
                                          (211111)  (41111)
                                                    (311111)
                                                    (2111111)
Also partitions of n + 1 without a fixed point or conjugate fixed point.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020. Mentions this sequence.

Cf. A001523 (stacks).
Cf. A006918, A252464, A325135, A325163, A325165.
Positive crank: A001522, ranked by A352874.
Zero crank: A064410, ranked by A342192.
Nonnegative crank: A064428, ranked by A352873.
Fixed point but no conjugate fixed point: A118199, ranked by A353316.
A000041 counts partitions, strict A000009.
A002467 counts permutations with a fixed point, complement A000166.
A115720/A115994 count partitions by Durfee square, rank statistic A257990.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points.
Cf. A000700, A088902, A114088, A300788, A330644, A352828, A352829, A352832.

a[n_]:=CoefficientList[Series[1+Sum[x^((k+1)^2)/Product[(1-x^i)^2,{i,1,k}],{k,0,n}],{x,0,n}],x]
(* second program *)
pml[ptn_]:=If[ptn=={},{},FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,ptn][[-3]]];
Table[Length[Select[IntegerPartitions[n],pml[#]=={1}&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

G.f.: 1 + sum(k>=0, x^((k+1)^2)/((1-x)^2*(1-x^2)^2*...*(1-x^k)^2)).

A117485 Expansion of x^9/((1-x)(1-x^2)(1-x^3))^2.

Original entry on oeis.org

1, 2, 5, 10, 18, 30, 49, 74, 110, 158, 221, 302, 407, 536, 698, 896, 1136, 1424, 1770, 2176, 2656, 3216, 3866, 4616, 5481, 6466, 7591, 8866, 10306, 11926, 13747, 15778, 18046, 20566, 23359, 26446, 29855, 33600, 37716, 42224, 47152, 52528, 58388, 64752, 71664

Offset: 9

Alford Arnold, Mar 22 2006

Molien series for S_3 X S_3, cf. A001399.
From Gus Wiseman, Apr 06 2019: (Start)
Also the number of integer partitions of n with Durfee square of length 3. The Heinz numbers of these partitions are given by A307386. For example, the a(9) = 1 through a(13) = 18 partitions are:
  (333)  (433)   (443)    (444)     (544)
         (3331)  (533)    (543)     (553)
                 (3332)   (633)     (643)
                 (4331)   (3333)    (733)
                 (33311)  (4332)    (4333)
                          (4431)    (4432)
                          (5331)    (4441)
                          (33321)   (5332)
                          (43311)   (5431)
                          (333111)  (6331)
                                    (33322)
                                    (33331)
                                    (43321)
                                    (44311)
                                    (53311)
                                    (333211)
                                    (433111)
                                    (3331111)
(End)

			As a cross-check, row sixteen of A115994 yields p(16) = 16 + 140 + 74 + 1.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).
Index entries for two-way infinite sequences

Column k=3 of A115994.
Cf. A000027 (for k=1), A006918 (for k=2), A117488, A117489, A001399, A117486.
Cf. A115720, A307386, A325164.

n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane, Mar 10 2007

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=3, stack): seq(count(subs(r=3, ZL), size=m), m=6..50) ; # Zerinvary Lajos, Jan 02 2008

CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3))^2,{x,0,50}],x] (* Harvey P. Dale, Oct 09 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==3&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Dec 12 2019

a(n) = floor((3*n^5 - 45*n^4 + 200*n^3 - 180*n^2 - 363*n + 1600)/12960 + n/27*(n%3==0) - n/32*(n%2==0)) \\ Hoang Xuan Thanh, Jul 17 2025

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>20. - Colin Barker, Dec 12 2019
From Hoang Xuan Thanh, May 17 2025: (Start)
a(n+3) = Sum_{x+2*y+3*z=n} x*y*z.
a(n+3) = n*(n^2-1)*(3*n^2-67)/12960 - floor((n+1)/3)/27 + [n mod 2 = 0]*n/32 + [n mod 3 = 0]*n/27 where [] is the Iverson bracket. (End)

Entry revised by N. J. A. Sloane, Mar 10 2007

A352826 Heinz numbers of integer partitions y without a fixed point y(i) = i. Such a fixed point is unique if it exists.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 24, 25, 26, 28, 29, 31, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 75, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 96, 97

Offset: 1

Gus Wiseman, Apr 06 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      1: ()          24: (2,1,1,1)     47: (15)
      3: (2)         25: (3,3)         48: (2,1,1,1,1)
      5: (3)         26: (6,1)         49: (4,4)
      6: (2,1)       28: (4,1,1)       50: (3,3,1)
      7: (4)         29: (10)          52: (6,1,1)
     10: (3,1)       31: (11)          53: (16)
     11: (5)         34: (7,1)         55: (5,3)
     12: (2,1,1)     35: (4,3)         56: (4,1,1,1)
     13: (6)         37: (12)          58: (10,1)
     14: (4,1)       38: (8,1)         59: (17)
     17: (7)         40: (3,1,1,1)     61: (18)
     19: (8)         41: (13)          62: (11,1)
     20: (3,1,1)     43: (14)          65: (6,3)
     22: (5,1)       44: (5,1,1)       67: (19)
     23: (9)         46: (9,1)         68: (7,1,1)

* = unproved
*These partitions are counted by A064428, strict A352828.
The complement is A352827.
The reverse version is A352830, counted by A238394.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point.
A008290 counts permutations by fixed points, nonfixed A098825.
A056239 adds up prime indices, row sums of A112798 and A296150.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824, A352825, A352831, A352832.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]==0&]

A352830 Numbers whose weakly increasing prime indices y have no fixed points y(i) = i.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141

Offset: 1

Gus Wiseman, Apr 06 2022

nonn

First differs from A325128 in lacking 75.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
All terms are odd.

			The terms together with their prime indices begin:
      1: {}        35: {3,4}     69: {2,9}     105: {2,3,4}
      3: {2}       37: {12}      71: {20}      107: {28}
      5: {3}       39: {2,6}     73: {21}      109: {29}
      7: {4}       41: {13}      77: {4,5}     111: {2,12}
     11: {5}       43: {14}      79: {22}      113: {30}
     13: {6}       47: {15}      83: {23}      115: {3,9}
     15: {2,3}     49: {4,4}     85: {3,7}     119: {4,7}
     17: {7}       51: {2,7}     87: {2,10}    121: {5,5}
     19: {8}       53: {16}      89: {24}      123: {2,13}
     21: {2,4}     55: {3,5}     91: {4,6}     127: {31}
     23: {9}       57: {2,8}     93: {2,11}    129: {2,14}
     25: {3,3}     59: {17}      95: {3,8}     131: {32}
     29: {10}      61: {18}      97: {25}      133: {4,8}
     31: {11}      65: {3,6}    101: {26}      137: {33}
     33: {2,5}     67: {19}     103: {27}      139: {34}

* = unproved
These partitions are counted by A238394, strict A025147.
These are the zeros of A352822.
*The reverse version is A352826, counted by A064428 (strict A352828).
*The complement reverse version is A352827, counted by A001522.
The complement is A352872, counted by A238395.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
A008290 counts permutations by fixed points, nonfixed A098825.
A056239 adds up prime indices, row sums of A112798 and A296150.
A114088 counts partitions by excedances.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824, A352825, A352831.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]==0&]

A115995 Sum of the sizes of the Durfee squares of all partitions of n.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 16, 23, 36, 52, 76, 106, 152, 207, 286, 386, 522, 691, 920, 1202, 1576, 2038, 2636, 3373, 4320, 5478, 6944, 8738, 10984, 13717, 17116, 21232, 26308, 32441, 39944, 48977, 59970, 73147, 89090, 108151, 131090, 158417, 191166, 230049, 276444

Offset: 0

Emeric Deutsch, Feb 11 2006

nonn

Also sum of positive cranks of all partitions of n, n>1; see A064391. - Vladeta Jovovic, Oct 20 2006

This sequence, its author and the author of the above comment were mentioned in the Andrews-Chan-Kim paper, where it is called C_1 (see the remark on page 6). - Omar E. Pol, Apr 06 2012

			a(4) = 6 because the partitions [4], [3,1], [2,2], [2,1,1] and [1,1,1,1] of 4 have Durfee squares of sizes 1,1,2,1 and 1, respectively.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Alois P. Heinz, Table of n, a(n) for n = 0..3000
George E. Andrews, Partitions and Durfee Dissection
George E. Andrews, Song Heng Chan, and Byungchan Kim, The odd moments of ranks and cranks
George E. Andrews, Frank G. Garvan, and Jie Liang, Self-conjugate vector partitions and the parity of the spt-function.
Atul Dixit, Bibekananda Maji, Partition implications of a new three parameter q-series identity, arXiv:1806.04424 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A115994, A115720, A115721, A115722.

g:= add(k*z^(k^2)/mul((1-z^j)^2,j=1..k),k=1..10): gser:=series(g,z=0,56): seq(coeff(gser,z,n), n=0..52);
# second Maple program:
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:
a:= n-> add(add(b(k, d)*b(n-d^2-k, d), k=0..n-d^2)*d, d=1..isqrt(n)):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012
# Third Maple program, based on Theorem 1 of Andrews-Chan-Kim:
M:=101;
qinf:=mul(1-q^i,i=1..M);
qinf:=series(qinf,q,M);
C1:=add((-1)^(n+1)*q^(n*(n+1)/2)/(1-q^n),n=1..M);
C1:=series(C1/qinf,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 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]]]] ; a[n_] := Sum[ Sum[b[k, d]*b[n - d^2 - k, d], {k, 0, n - d^2}]*d, {d, 1, Sqrt[n]}]; Table [a[n], {n, 0, 70}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)

N=66; x='x+O('x^N); concat([0], Vec( sum(n=0,N, n*x^(n^2) / prod(k=1,n, 1-x^k)^2))) \\ Joerg Arndt, Mar 26 2014

[sum(p.frobenius_rank() for p in Partitions(n)) for n in range(45)] # Peter Luschny, Sep 15 2014

G.f.: Sum_{k>=1} (k*z^(k^2) / Product_{j=1..k} (1 - z^j)^2 ).
a(n) = Sum_{k=1..floor(sqrt(n))} k*A115994(n,k).
Convolution of A067742 and A000041. - Vladeta Jovovic, Oct 20 2006
a(n) = A195012(n) + A209616(n), n >= 1. - Omar E. Pol, Apr 06 2012
a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Jan 02 2019

Edited and verified by Franklin T. Adams-Watters, Mar 11 2006

A352872 Numbers whose weakly increasing prime indices y have a fixed point y(i) = i.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114

Offset: 1

Gus Wiseman, Apr 06 2022

nonn

First differs from A118672 in having 75.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
      2: {1}           28: {1,1,4}         56: {1,1,1,4}
      4: {1,1}         30: {1,2,3}         58: {1,10}
      6: {1,2}         32: {1,1,1,1,1}     60: {1,1,2,3}
      8: {1,1,1}       34: {1,7}           62: {1,11}
      9: {2,2}         36: {1,1,2,2}       63: {2,2,4}
     10: {1,3}         38: {1,8}           64: {1,1,1,1,1,1}
     12: {1,1,2}       40: {1,1,1,3}       66: {1,2,5}
     14: {1,4}         42: {1,2,4}         68: {1,1,7}
     16: {1,1,1,1}     44: {1,1,5}         70: {1,3,4}
     18: {1,2,2}       45: {2,2,3}         72: {1,1,1,2,2}
     20: {1,1,3}       46: {1,9}           74: {1,12}
     22: {1,5}         48: {1,1,1,1,2}     75: {2,3,3}
     24: {1,1,1,2}     50: {1,3,3}         76: {1,1,8}
     26: {1,6}         52: {1,1,6}         78: {1,2,6}
     27: {2,2,2}       54: {1,2,2,2}       80: {1,1,1,1,3}
For example, the multiset {2,3,3} with Heinz number 75 has a fixed point at position 3, so 75 is in the sequence.

* = unproved
These partitions are counted by A238395, strict A096765.
These are the nonzero positions in A352822.
*The complement reverse version is A352826, counted by A064428.
*The reverse version is A352827, counted by A001522 (strict A352829).
The complement is A352830, counted by A238394 (strict A025147).
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
A008290 counts permutations by fixed points, nonfixed A098825.
A056239 adds up prime indices, row sums of A112798 and A296150.
A114088 counts partitions by excedances.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A093641, A257990, A325187, A342192, A352486, A352823, A352824, A352825, A352831, A352832.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>0&]

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

A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 26, 32, 38, 46, 56, 66, 78, 92, 106, 123, 142, 162, 186, 214, 244, 280, 322, 368, 422, 484, 552, 630, 718, 815, 924, 1046, 1180, 1330, 1498, 1682, 1888, 2118, 2372, 2656, 2972, 3322, 3712, 4146, 4626

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 1 through a(12) = 12 partitions (A-C = 10..12; empty column indicated by dot; 0 is the empty partition):
   0  .  2  3    4    5    6    7    8     9     A      B      C
            21   31   41   51   43   53    54    64     65     75
                                61   71    63    73     74     84
                                     431   81    91     83     93
                                           432   532    A1     B1
                                           531   541    542    642
                                                 631    632    651
                                                 4321   641    732
                                                        731    741
                                                        5321   831
                                                               5421
                                                               6321

Alois P. Heinz, Table of n, a(n) for n = 0..5000

The version for permutations is A000166, complement A002467.
The reverse version is A025147, complement A238395, non-strict A238394.
The non-strict version is A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A238351, complement A352875.
The complement is A352829, non-strict A001522 (unproved, ranked by A352827 or A352874).
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352824, A352825, A352830, A352872.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]==0&]],{n,0,30}]

G.f.: Sum_{n>=0} q^(n*(3*n+1)/2)*Product_{k=1..n} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

A352833 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k fixed points, k = 0, 1.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 5, 8, 7, 12, 10, 16, 14, 23, 19, 30, 26, 42, 35, 54, 47, 73, 62, 94, 82, 124, 107, 158, 139, 206, 179, 260, 230, 334, 293, 420, 372, 532, 470, 664, 591, 835, 740, 1034, 924, 1288, 1148, 1588, 1422, 1962, 1756, 2404, 2161

Offset: 0

Gus Wiseman, Apr 08 2022

A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists, so all columns k > 1 are zeros.
Conjecture:
(1) This is A064428 interleaved with A001522.
(2) Reversing rows gives A300788, the strict version of A300787.

			Triangle begins:
  0: {1,0}
  1: {0,1}
  2: {1,1}
  3: {2,1}
  4: {3,2}
  5: {4,3}
  6: {6,5}
  7: {8,7}
  8: {12,10}
  9: {16,14}
For example, row n = 7 counts the following partitions:
  (7)       (52)
  (61)      (421)
  (511)     (322)
  (43)      (3211)
  (4111)    (2221)
  (331)     (22111)
  (31111)   (1111111)
  (211111)

Row sums are A000041.
The version for permutations is A008290, for nonfixed points A098825.
The columns appear to be A064428 and A001522.
The version counting strong nonexcedances is A114088.
The version for compositions is A238349, rank statistic A352512.
The version for reversed partitions is A238352.
Reversing rows appears to give A300788, the strict case of A300787.
A000700 counts self-conjugate partitions, ranked by A088902.
A115720 and A115994 count partitions by their Durfee square.
A330644 counts non-self-conjugate partitions, ranked by A352486.
Cf. A000701, A219282, A257990, A350839, A352513, A352521-A352525.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],pq[#]==k&]],{n,0,15},{k,0,1}]

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

cp's OEIS Frontend

A188674 Stack polyominoes with square core.

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A117485 Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2.

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A352826 Heinz numbers of integer partitions y without a fixed point y(i) = i. Such a fixed point is unique if it exists.

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A352830 Numbers whose weakly increasing prime indices y have no fixed points y(i) = i.

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A115995 Sum of the sizes of the Durfee squares of all partitions of n.

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A352872 Numbers whose weakly increasing prime indices y have a fixed point y(i) = i.

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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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A117485 Expansion of x^9/((1-x)(1-x^2)(1-x^3))^2.