A188674 -id:A188674 - cp's OEIS frontend

A352873 Heinz numbers of integer partitions with nonnegative crank, counted by A064428.

1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90

Offset: 1

Gus Wiseman, Apr 09 2022

nonn

First differs from A042968, A059557, and A195291 in lacking 2 and having 100.
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 crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's). [Definition copied from A342192; see A064428 for a different wording.]

			The terms together with their prime indices begin:
     1: ()         22: (5,1)      42: (4,2,1)
     3: (2)        23: (9)        43: (14)
     5: (3)        25: (3,3)      45: (3,2,2)
     6: (2,1)      26: (6,1)      46: (9,1)
     7: (4)        27: (2,2,2)    47: (15)
     9: (2,2)      29: (10)       49: (4,4)
    10: (3,1)      30: (3,2,1)    50: (3,3,1)
    11: (5)        31: (11)       51: (7,2)
    13: (6)        33: (5,2)      53: (16)
    14: (4,1)      34: (7,1)      54: (2,2,2,1)
    15: (3,2)      35: (4,3)      55: (5,3)
    17: (7)        37: (12)       57: (8,2)
    18: (2,2,1)    38: (8,1)      58: (10,1)
    19: (8)        39: (6,2)      59: (17)
    21: (4,2)      41: (13)       61: (18)

* = unproved
These partitions are counted by A064428.
The case of zero crank is A342192, counted by A064410.
The case of positive crank is A352874.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
Cf. A065770, A093641, A118199, A188674, A252464, A257990, A325163, A325169, A344609, A352828, A352831.

ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Select[Range[100],ck[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]>=0&]

Union of A352874 and A342192.

A352874 Heinz numbers of integer partitions with positive crank, counted by A001522.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101, 102, 103, 105, 107, 109

Offset: 1

Gus Wiseman, Apr 09 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 crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's). [Definition copied from A342192; see A064428 for a different wording.]

			The terms together with their prime indices begin:
      3: (2)         30: (3,2,1)     54: (2,2,2,1)
      5: (3)         31: (11)        55: (5,3)
      7: (4)         33: (5,2)       57: (8,2)
      9: (2,2)       35: (4,3)       59: (17)
     11: (5)         37: (12)        61: (18)
     13: (6)         39: (6,2)       63: (4,2,2)
     15: (3,2)       41: (13)        65: (6,3)
     17: (7)         42: (4,2,1)     66: (5,2,1)
     18: (2,2,1)     43: (14)        67: (19)
     19: (8)         45: (3,2,2)     69: (9,2)
     21: (4,2)       47: (15)        70: (4,3,1)
     23: (9)         49: (4,4)       71: (20)
     25: (3,3)       50: (3,3,1)     73: (21)
     27: (2,2,2)     51: (7,2)       75: (3,3,2)
     29: (10)        53: (16)        77: (5,4)

* = unproved
These partitions are counted by A001522.
The case of zero crank is A342192, counted by A064410.
The case of nonnegative crank is A352873, counted by A064428.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
Cf. A065770, A093641, A118199, A188674, A252464, A257990, A325163, A325169, A344609, A352828, A352831.

ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Select[Range[100],ck[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]>0&]

Complement of A342192 in A352873.

A325165 Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4, 0, 2, 0, 0, 0, 5, 0, 3, 2, 0, 0, 0, 6, 0, 4, 4, 0, 0, 0, 0, 7, 0, 5, 6, 3, 0, 0, 0, 0, 8, 0, 7, 8, 6, 0, 0, 0, 0, 0, 9, 0, 9, 10, 9, 4, 0, 0, 0, 0, 0, 10, 0, 13, 12, 12, 8, 0, 0, 0, 0, 0, 0, 11

Offset: 0

Gus Wiseman, Apr 05 2019

The k-th part of the inner lining partition of an integer partition is the number of squares in its Young diagram that are k diagonal steps 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
which has diagonal distances from the lower-right boundary equal to
  3 3 3 2 1 1
  3 2 2 2 1
  2 2 1 1 1
  1 1 1
so the inner lining sequence is (9,6,4) with last part 4, so (6,5,5,3) is counted under T(19,4).

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  0  3
  0  1  0  0  4
  0  2  0  0  0  5
  0  3  2  0  0  0  6
  0  4  4  0  0  0  0  7
  0  5  6  3  0  0  0  0  8
  0  7  8  6  0  0  0  0  0  9
  0  9 10  9  4  0  0  0  0  0 10
  0 13 12 12  8  0  0  0  0  0  0 11
  0 17 16 15 12  5  0  0  0  0  0  0 12
  0 24 20 18 16 10  0  0  0  0  0  0  0 13
  0 31 28 21 20 15  6  0  0  0  0  0  0  0 14
  0 42 36 27 24 20 12  0  0  0  0  0  0  0  0 15
  0 54 50 33 28 25 18  7  0  0  0  0  0  0  0  0 16
  0 71 64 45 32 30 24 14  0  0  0  0  0  0  0  0  0 17
  0 90 86 57 40 35 30 21  8  0  0  0  0  0  0  0  0  0 18
Row n = 9 counts the following partitions (empty columns not shown):
  (72)       (63)      (54)     (9)
  (333)      (522)     (432)    (81)
  (621)      (531)     (441)    (711)
  (5211)     (4221)    (3222)   (6111)
  (42111)    (4311)    (3321)   (51111)
  (321111)   (32211)   (22221)  (411111)
  (2211111)  (33111)            (3111111)
             (222111)           (21111111)
                                (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000041. Column k = 1 is A188674.

Cf. A006918, A115720, A115994, A117485, A252464, A257990, A325163, A325166, A325168.

pml[ptn_]:=If[ptn=={},{},FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,ptn][[-3]]];
Table[Length[Select[IntegerPartitions[n],Total[pml[#]]==k&]],{n,0,10},{k,0,n}]

T(n) = {my(v=Vec(1+sum(k=1, sqrtint(n), x^(k^2)/((1-y*x^k)*prod(j=1, k-1, 1 - x^j + O(x^(n+1-k^2))))^2))); vector(#v, i, Vecrev(v[i], -i))}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 19 2023

G.f.: A(x,y) = 1 + Sum_{k>=1} x^(k^2)/((1 - y*x^k) * Product_{j=1..k-1} (1 - x^j))^2. - Andrew Howroyd, Jan 19 2023

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]]&]

A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 1, 0, 1, 4, 2, 0, 0, 2, 5, 3, 2, 0, 2, 3, 6, 4, 4, 0, 0, 4, 4, 7, 5, 6, 3, 0, 3, 6, 5, 8, 7, 8, 7, 0, 1, 6, 8, 6, 9, 9, 10, 11, 4, 0, 6, 9, 10, 7, 10, 13, 12, 15, 10, 0, 2, 11, 12, 12, 8, 11

Offset: 1

Omar E. Pol, Dec 12 2019

This triangle has the property that it contains the triangle A049597, since if we replace with zeros the positive terms before the first zero in the row n of this triangle, we get the triangle A049597.
Hence the sum of the terms after the last zero in row n equals A000041(n), the number of partitions of n (see the Example section).
Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).

			Triangle begins:
   1;
   0,  2;
   0,  0,  3;
   1,  0,  1,  4;
   2,  0,  0,  2,  5;
   3,  2,  0,  2,  3,  6;
   4,  4,  0,  0,  4,  4,  7;
   5,  6,  3,  0,  3,  6,  5,  8;
   7,  8,  7,  0,  1,  6,  8,  6,  9;
   9, 10, 11,  4,  0,  6,  9, 10,  7, 10;
  13, 12, 15, 10,  0,  2, 11, 12, 12,  8, 11;
Figure 1 below shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. Figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.
.
.                                     Right-angles   Right
Part   Ferrers diagram         Part   diagram        angle
                                      _ _ _ _ _ _ _
  7    * * * * * * *             7   |  _ _ _ _ _ _|  14
  6    * * * * * *               6   | |  _ _ _ _|     8
  3    * * *                     3   | | | |           2
  3    * * *                     3   | | |_|
  2    * *                       2   | |_|
  1    *                         1   | |
  1    *                         1   | |
  1    *                         1   |_|
.
       Figure 1.                      Figure 2.
.
For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:
.
    _       _ _       _ _ _       _ _ _ _       _ _ _ _ _
  1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1|_|
  1| |    1| |      1| |        1|_|
  1| |    1| |      1|_|
  1| |    1|_|
  1|_|
    _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _
  6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8
  1| |              1|_|
  1|_|
.
    _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _
  2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7
  2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1
  1| |      1| |        1| |          1|_|
  1| |      1| |        1|_|
  1| |      1|_|
  1|_|
.
    _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _
  2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6
  2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2
  2| |_|    2| |_|      1| |        2|_|_|        1|_|
  1| |      1|_|        1|_|
  1|_|
.
    _ _       _ _ _        _ _ _ _
  2|  _|5   3|  _ _|5    4|  _ _ _|5
  2| | |3   3| |  _|3    4|_|_ _ _|3
  2| | |    2|_|_|
  2|_|_|
.
There are  5 right angles of size 1, so T(8,1) = 5.
There are  6 right angles of size 2, so T(8,2) = 6.
There are  3 right angles of size 3, so T(8,3) = 3.
There are no right angle  of size 4, so T(8,4) = 0.
There are  3 right angles of size 5, so T(8,5) = 3.
There are  6 right angles of size 6, so T(8,6) = 6.
There are  5 right angles of size 7, so T(8,7) = 5.
There are  8 right angles of size 8, so T(8,8) = 8.
Hence the 8th row of triangle is [5, 6, 3, 0, 3, 6, 5, 8].
Note that the sum of the terms after the last zero is 3 + 6 + 5 + 8 = 22, equaling A000041(8) = 22, the number of partitions of 8.

G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143 [Defines the right angles in the Ferrers graph of a partition. - N. J. A. Sloane, Nov 20 2020]

Row sums give A115995.
Right border gives A000027.
Cf. A000041, A049597, A083480, A083906, A188674, A259481, A325458, A330375, A330378, A330379.

A295341 The number of partitions of n in which at least one part is a multiple of 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 6, 9, 14, 20, 29, 41, 57, 78, 106, 142, 189, 250, 327, 425, 549, 705, 900, 1144, 1445, 1819, 2279, 2844, 3534, 4379, 5403, 6648, 8152, 9969, 12152, 14780, 17920, 21682, 26163, 31504, 37842, 45371, 54270, 64800, 77211, 91842, 109031, 129235, 152897

Offset: 0

R. J. Mathar, Nov 20 2017

nonn

From Gus Wiseman, May 23 2022: (Start)
Also the number of integer partitions of n with at least one part appearing more than twice. The Heinz numbers of these partitions are given by A046099. For example, the a(0) = 0 though a(8) = 9 partitions are:
  .  .  .  (111)  (1111)  (2111)   (222)     (2221)     (2222)
                          (11111)  (3111)    (4111)     (5111)
                                   (21111)   (22111)    (22211)
                                   (111111)  (31111)    (32111)
                                             (211111)   (41111)
                                             (1111111)  (221111)
                                                        (311111)
                                                        (2111111)
                                                        (11111111)
(End)

			From _Gus Wiseman_, May 23 2022: (Start)
The a(0) = 0 through a(8) = 9 partitions with a part that is a multiple of 3:
  .  .  .  (3)  (31)  (32)   (6)     (43)     (53)
                      (311)  (33)    (61)     (62)
                             (321)   (322)    (332)
                             (3111)  (331)    (431)
                                     (3211)   (611)
                                     (31111)  (3221)
                                              (3311)
                                              (32111)
                                              (311111)
(End)

The complement is counted by A000726, ranked by A004709.
These partitions are ranked by A354235.
This is column k = 3 of A354234.
For 2 instead of 3 we have A047967, ranked by A013929 and A324929.
For 4 instead of 3 we have A295342, ranked by A046101.
A000041 counts integer partitions, strict A000009.
A046099 lists non-cubefree numbers.
Cf. A001522, A006918, A064410, A064428, A117485, A188674, A325187, A325534.

Table[Length[Select[IntegerPartitions[n],MemberQ[#/3,?IntegerQ]&]],{n,0,30}] (* _Gus Wiseman, May 23 2022 *)
Table[Length[Select[IntegerPartitions[n],MatchQ[#,{_,x_,x_,x_,_}]&]],{n,0,30}] (* Gus Wiseman, May 23 2022 *)

a(n) = A000041(n)-A000726(n).

A226541 Number of unimodal compositions of n where the maximal part appears three times.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 3, 5, 7, 11, 16, 24, 34, 51, 71, 102, 143, 201, 276, 384, 522, 714, 964, 1301, 1739, 2328, 3084, 4085, 5377, 7064, 9226, 12036, 15616, 20228, 26092, 33584, 43067, 55125, 70308, 89502, 113598, 143889, 181755, 229160, 288186, 361750, 453046, 566346, 706464

Offset: 0

Joerg Arndt, Jun 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A006330 (max part appears once), A114921 (max part appears twice).
Cf. A188674 (max part m appears m times), A001522 (max part m appears at least m times).
Cf. A001523 (max part appears any number of times).
Cf. A000009 (symmetric, max part m appears once; also symmetric, max part appears an odd number of times).
Cf. A035363 (symmetric, max part m appears twice; also symmetric, max part appears an even number of times).
Cf. A087897 (symmetric, max part m appears 3 times).
Cf. A027349 (symmetric, max part m appears m times), A189357 (symmetric, max part m appears at least m times).
Column k=3 of A247255.

N=66; x='x+O('x^N); Vec(sum(n=0,N, x^(3*n) / prod(k=1,n-1, 1-x^k )^2 ))

G.f.: sum(n>=0, x^(3*n) / prod(k=1..n-1, 1-x^k )^2 ); replace 3 by m to obtain g.f. for "... max part appears m times".

a(n) ~ Pi^2 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Oct 24 2018

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}]

A330379 Triangle read by rows: T(n,k) (1 <= k <= n) is the sum of the sizes of all right angles of size k of all partitions of n.

Original entry on oeis.org

1, 0, 4, 0, 0, 9, 1, 0, 3, 16, 2, 0, 0, 8, 25, 3, 4, 0, 8, 15, 36, 4, 8, 0, 0, 20, 24, 49, 5, 12, 9, 0, 15, 36, 35, 64, 7, 16, 21, 0, 5, 36, 56, 48, 81, 9, 20, 33, 16, 0, 36, 63, 80, 63, 100, 13, 24, 45, 40, 0, 12, 77, 96, 108, 80, 121

Offset: 1

Omar E. Pol, Dec 31 2019

Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).

			Triangle begins:
   1;
   0,  4;
   0,  0,  9;
   1,  0,  3, 16;
   2,  0,  0,  8, 25;
   3,  4,  0,  8, 15, 36;
   4,  8,  0,  0, 20, 24, 49;
   5, 12,  9,  0, 15, 36, 35, 64;
   7, 16, 21,  0,  5, 36, 56, 48,  81;
   9, 20, 33, 16,  0, 36, 63, 80,  63, 100;
  13, 24, 45, 40,  0, 12, 77, 96, 108,  80, 121;
...
Below the figure 1 shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. The figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.
.
.                                     Right-angles   Right
Part   Ferrers diagram         Part   diagram        angle
                                      _ _ _ _ _ _ _
  7    * * * * * * *             7   |  _ _ _ _ _ _|  14
  6    * * * * * *               6   | |  _ _ _ _|     8
  3    * * *                     3   | | | |           2
  3    * * *                     3   | | |_|
  2    * *                       2   | |_|
  1    *                         1   | |
  1    *                         1   | |
  1    *                         1   |_|
.
       Figure 1.                      Figure 2.
.
For n = 8 the partitions of 8 and their respective right-angles diagrams look as shown below:
.
    _       _ _       _ _ _       _ _ _ _       _ _ _ _ _
  1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1| |
  1| |    1| |      1| |        1| |          1|_|
  1| |    1| |      1| |        1|_|
  1| |    1| |      1|_|
  1| |    1|_|
  1|_|
    _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _
  6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8
  1| |              1|_|
  1|_|
.
    _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _
  2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7
  2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1
  1| |      1| |        1| |          1|_|
  1| |      1| |        1|_|
  1| |      1|_|
  1|_|
.
    _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _
  2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6
  2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2
  2| |_|    2| |_|      1| |        2|_|_|        1|_|
  1| |      1|_|        1|_|
  1|_|
.
    _ _       _ _ _        _ _ _ _
  2|  _|5   3|  _ _|5    4|  _ _ _|5
  2| | |3   3| |  _|3    4|_|_ _ _|3
  2| | |    2|_|_|
  2|_|_|
.
There are  5 right angles of size 1, so T(8,1) = 5*1 = 5.
There are  6 right angles of size 2, so T(8,2) = 6*2 = 12.
There are  3 right angles of size 3, so T(8,3) = 3*3 = 9.
There are no right angle  of size 4, so T(8,4) = 0*4 = 0.
There are  3 right angles of size 5, so T(8,5) = 3*5 = 15.
There are  6 right angles of size 6, so T(8,6) = 6*6 = 36.
There are  5 right angles of size 7, so T(8,7) = 5*7 = 35.
There are  8 right angles of size 8, so T(8,8) = 8*8 = 64.
Hence the 8th row of triangle is [5, 12, 9, 0, 15, 36, 35, 64].
The row sum gives A066186(8) = 8*A000041(8) = 8*22 = 176.

G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143.

Row sums give A066186, n >= 1.
Row sums of the terms that are after last zero give A179862.
Cf. A188674.
Cf. A000041, A000290, A330369, A330375, A330378.

T(n,k) = k*A330369(n,k).

A352875 Number of integer compositions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 1, 2, 5, 10, 21, 42, 86, 174, 351, 708, 1424, 2861, 5743, 11520, 23092, 46269, 92673, 185562, 371469, 743491, 1487870, 2977164, 5956616, 11916910, 23839736, 47688994, 95393322, 190811346, 381662507, 763389209, 1526881959, 3053930971, 6108131542, 12216698288

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 0 through a(5) = 10 compositions (empty column indicated by dot):
  .  (1)  (11)  (12)   (13)    (14)
                (111)  (22)    (32)
                       (112)   (113)
                       (121)   (122)
                       (1111)  (131)
                               (221)
                               (1112)
                               (1121)
                               (1211)
                               (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for partitions is A001522, ranked by A352827 (unproved).
The version for permutations is A002467, complement A000166.
The complement for partitions is A064428, ranked by A352826 (unproved).
This is the sum of latter columns of A238349, nonfixed A352523.
The complement is counted by A238351.
The complement for reversed partitions is A238394, ranked by A352830.
The version for reversed partitions is A238395, ranked by A352872.
The case of just one fixed point is A240736.
A008290 counts permutations by fixed points, nonfixed A098825.
A011782 counts compositions.
A115720 and A115994 count partitions by Durfee square.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352512 counts fixed points in standard compositions, nonfixed A352513.
A352521 = comps by subdiags, first col A219282, rank stat A352514.
A352522 = comps by weak subdiags, first col A238874, rank stat A352515.
A352524 = comps by superdiags, first col A008930, rank stat A352516.
A352525 = comps by weak superdiags, col k=1 A177510, rank stat A352517.
A352833 counts partitions by fixed points.
Cf. A006918, A008292, A088218, A114088, A123125, A188674, A257990.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pq[#]>0&]],{n,0,15}]

S(v,u,c)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
seq(n)={my(v=vector(1+n), s=vector(#v, i, 2^(i-2))); v[1]=1; s[1]=0; for(i=1, n, v=S(v, vector(n, j, if(j==i,'x,1)), O(x)); s-=apply(p->polcoef(p,0), v)); s} \\ Andrew Howroyd, Jan 02 2023

a(n) = 2^(n-1) - A238351(n) for n >= 1. - Andrew Howroyd, Jan 02 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2023

cp's OEIS Frontend

A352873 Heinz numbers of integer partitions with nonnegative crank, counted by A064428.

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A352874 Heinz numbers of integer partitions with positive crank, counted by A001522.

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A325165 Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.

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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.

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A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n.

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A295341 The number of partitions of n in which at least one part is a multiple of 3.

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A226541 Number of unimodal compositions of n where the maximal part appears three times.

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

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