A238352 -id:A238352 - cp's OEIS frontend

A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.

1, 1, 1, 1, 2, 2, 3, 5, 6, 9, 1, 11, 18, 3, 21, 35, 8, 41, 67, 20, 80, 131, 44, 1, 157, 257, 94, 4, 310, 505, 197, 12, 614, 996, 406, 32, 1218, 1973, 825, 80, 2421, 3915, 1669, 186, 1, 4819, 7781, 3364, 415, 5, 9602, 15486, 6762, 901, 17, 19147, 30855, 13567, 1918, 49

Offset: 0

Gus Wiseman, Mar 22 2022

			Triangle begins:
     1
     1
     1     1
     2     2
     3     5
     6     9     1
    11    18     3
    21    35     8
    41    67    20
    80   131    44     1
   157   257    94     4
   310   505   197    12
   614   996   406    32
For example, row n = 5 counts the following compositions:
  (113)    (5)     (23)
  (122)    (14)
  (1112)   (32)
  (1121)   (41)
  (1211)   (131)
  (11111)  (212)
           (221)
           (311)
           (2111)

Andrew Howroyd, Table of n, a(n) for n = 0..2507 (rows 0..200)
MathOverflow, Why 'excedances' of permutations? [closed].

The version for permutations is A008292, weak A123125.
Column k = 0 is A008930.
Row sums are A011782.
The opposite version for partitions is A114088.
The weak version for partitions is A115994.
Column k = 1 is A351983.
The corresponding rank statistic is A352516.
The opposite version is A352521, first col A219282, rank statistic A352514.
The weak opposite version is A352522, first col A238874, rank stat A352515.
The weak version is A352525, first col (k = 1) A177510, rank stat A352517.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352487 lists the excedance set of A122111, opposite A352490.
A352523 counts comps by unfixed points, first A352520, rank stat A352513.
Cf. A088218, A098825, A238352, A350839.

pd[y_]:=Length[Select[Range[Length[y]],#

S(v,u)={vector(#v, k, sum(i=1, k-1, v[k-i]*u[i]))}
T(n)={my(v=vector(1+n), s); v[1]=1; s=v; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1))); s+=v); [Vecrev(p) | p<-s]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 02 2023

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}]

A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 3, 4, 3, 4, 4, 5, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2

Offset: 0

Gus Wiseman, Mar 22 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 83rd composition in standard order is (2,3,1,1), with strong nonexcedances {3,4}, so a(83) = 2.

MathOverflow, Why 'excedances' of permutations? [closed].

Positions of first appearances are A000225.
The weak version is A352515, counted by A352522 (first column A238874).
The opposite version is A352516, counted by A352524 (first column A008930).
The weak opposite version is A352517, counted by A352525 (first A177510).
The triangle A352521 counts these compositions (first column A219282).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed parts, first col A238351, rank stat A352512.
A352490 is the (strong) nonexcedance set of A122111.
A352523 counts comps by unfixed parts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pa[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
Table[pa[stc[n]],{n,0,30}]

A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 3, 4, 3, 4, 4, 5, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 5, 2, 4, 4, 5, 4, 5, 5, 6, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 2, 3, 3

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 89th composition in standard order is (2,1,3,1), with weak nonexcedances {2,3,4}, so a(89) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positions of first appearances are A000225.
The strong version is A352514, counted by A352521 (first column A219282).
The strong opposite version is A352516, counted by A352524 (first A008930).
The opposite version is A352517, counted by A352525 (first column A177510).
Triangle A352522 counts these comps (first col A238874), partitions A115994.
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352488 is the weak nonexcedance set of A122111.
A352523 counts comps by unfixed pts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A131577, A238352.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
paw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[paw[stc[n]],{n,0,30}]

A352516 Number of excedances (parts above the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 5392th composition in standard order is (2,2,4,5), with excedances {1,3,4}, so a(5392) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positions of first appearances are A104462.
The opposite version is A352514, counted by A352521 (first column A219282).
The weak opposite version is A352515, counted by A352522 (first A238874).
The weak version is A352517, counted by A352525 (first column A177510).
The triangle A352524 counts these compositions (first column A008930).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352487 is the excedance set of A122111.
A352523 counts comps by unfixed points, first A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd[y_]:=Length[Select[Range[Length[y]],#

A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 169th composition in standard order is (2,2,3,1), with weak excedances {1,2,3}, so a(169) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positive positions of first appearances are A164894.
The version for partitions is A257990.
The strong opposite version is A352514, counted by A352521 (first A219282).
The opposite version is A352515, counted by A352522 (first column A238874).
The strong version is A352516, counted by A352524 (first column A008930).
The triangle A352525 counts these compositions (first column A177510).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352489 is the weak excedance set of A122111.
A352523 counts comps by unfixed points, first A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
Table[pdw[stc[n]],{n,0,30}]

A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 26, 30, 36, 42, 50, 60, 70, 82, 96, 110, 126, 144, 163, 184, 208, 234, 264, 298, 336, 380, 430, 486, 550, 622, 702, 792, 892, 1002, 1125, 1260, 1408, 1572, 1752, 1950, 2168, 2408, 2672

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(11) = 2 through a(17) = 12 partitions (A-F = 10..15):
  (92)   (A2)   (B2)    (C2)    (D2)     (E2)     (F2)
  (821)  (543)  (643)   (653)   (753)    (763)    (863)
         (921)  (A21)   (743)   (843)    (853)    (953)
                (5431)  (B21)   (C21)    (943)    (A43)
                        (5432)  (6432)   (D21)    (E21)
                        (6431)  (6531)   (6532)   (7532)
                                (7431)   (7432)   (7631)
                                (54321)  (7531)   (8432)
                                         (8431)   (8531)
                                         (64321)  (9431)
                                                  (65321)
                                                  (74321)

The non-strict version is A001522 (unproved, ranked by A352827 or A352874).
The version for permutations is A002467, complement A000166.
The reverse version is A096765 (or A025147 shifted right once).
The non-strict reverse version is A238395, ranked by A352872.
The complement is counted by A352828, non-strict A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A352875, complement A238351.
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.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352823, A352824, A352825, A352832.

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

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

A352831 Numbers whose weakly increasing prime indices y have exactly one fixed point y(i) = i.

Original entry on oeis.org

2, 4, 8, 9, 10, 12, 14, 16, 22, 24, 26, 27, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 58, 60, 62, 63, 64, 68, 70, 72, 74, 75, 76, 80, 81, 82, 86, 88, 92, 94, 96, 98, 99, 104, 106, 108, 110, 112, 116, 117, 118, 120, 122, 124, 125, 128, 130, 132, 134, 135, 136

Offset: 1

Gus Wiseman, Apr 08 2022

nonn

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}             36: {1,1,2,2}         74: {1,12}
      4: {1,1}           38: {1,8}             75: {2,3,3}
      8: {1,1,1}         40: {1,1,1,3}         76: {1,1,8}
      9: {2,2}           44: {1,1,5}           80: {1,1,1,1,3}
     10: {1,3}           46: {1,9}             81: {2,2,2,2}
     12: {1,1,2}         48: {1,1,1,1,2}       82: {1,13}
     14: {1,4}           52: {1,1,6}           86: {1,14}
     16: {1,1,1,1}       58: {1,10}            88: {1,1,1,5}
     22: {1,5}           60: {1,1,2,3}         92: {1,1,9}
     24: {1,1,1,2}       62: {1,11}            94: {1,15}
     26: {1,6}           63: {2,2,4}           96: {1,1,1,1,1,2}
     27: {2,2,2}         64: {1,1,1,1,1,1}     98: {1,4,4}
     28: {1,1,4}         68: {1,1,7}           99: {2,2,5}
     32: {1,1,1,1,1}     70: {1,3,4}          104: {1,1,1,6}
     34: {1,7}           72: {1,1,1,2,2}      106: {1,16}
For example, 63 is in the sequence because its prime indices {2,2,4} have a unique fixed point at the second position.

* = unproved
These are the positions of 1's in A352822.
*The reverse version for no fixed points is A352826, counted by A064428.
*The reverse version is A352827, counted by A001522 (strict A352829).
The version for no fixed points is A352830, counted by A238394.
These partitions are counted by A352832, compositions A240736.
Allowing more than one fixed point gives 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.
A115720 and A115994 count partitions by their Durfee square.
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, A177510, A257990, A342192, A349158, A351983, A352520, A352823, A352824.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]==1&]

A118199 Number of partitions of n having no parts equal to the size of their Durfee squares.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 40, 53, 68, 89, 113, 146, 184, 234, 293, 369, 458, 572, 706, 874, 1073, 1320, 1611, 1970, 2393, 2909, 3518, 4255, 5122, 6167, 7394, 8862, 10585, 12637, 15038, 17886, 21213, 25141, 29723, 35112, 41383, 48737, 57278

Offset: 0

Emeric Deutsch, Apr 14 2006

nonn

a(n) = A118198(n,0).
From Gus Wiseman, May 21 2022: (Start)
Also the number of integer partitions of n > 0 that have a fixed point but whose conjugate does not, ranked by A353316. For example, the a(5) = 1 through a(10) = 10 partitions are:
  11111   222      322       422        522         622
          111111   2221      2222       3222        4222
                   1111111   3221       4221        5221
                             22211      22221       22222
                             11111111   32211       32221
                                        222111      42211
                                        111111111   222211
                                                    322111
                                                    2221111
                                                    1111111111
Partitions w/ a fixed point: A001522 (unproved), ranked by A352827 (cf. A352874).
Partitions w/o a fixed point: A064428 (unproved), ranked by A352826 (cf. A352873).
Partitions w/ a fixed point and a conjugate fixed point: A188674, reverse A325187, ranked by A353317.
Partitions w/o a fixed point or conjugate fixed point: A188674 (shifted).
(End)

			a(7) = 3 because we have [7] with size of Durfee square 1, [4,3] with size of Durfee square 2 and [3,3,1] with size of Durfee square 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Column k=0 of A118198.
A000041 counts partitions, strict A000009.
A000700 = self-conjugate partitions, ranked by A088902, complement A330644.
A002467 counts permutations with a fixed point, complement A000166.
A064410 counts partitions of crank 0, ranked by A342192.
A115720 and A115994 count partitions by Durfee square, rank stat 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. A114088, A300788, A325039, A350839, A352828, A352829, A352832.

g:=1+sum(x^(k^2+k)/(1-x^k)/product((1-x^i)^2,i=1..k-1),k=1..20): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=0..54);
# 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*(d+1)-k, d-1),
                k=0..n-d*(d+1)), d=0..floor(sqrt(n))):
seq(a(n), n=0..70);  # 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]]]]; a[n_] := Sum[Sum[b[k, d]*b[n-d*(d+1)-k, d-1], {k, 0, n-d*(d+1)}], {d, 0, Floor[Sqrt[n]]}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],pq[#]>0&&pq[conj[#]]==0&]],{n,0,30}] (* a(0) = 0, Gus Wiseman, May 21 2022 *)

G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - Vaclav Kotesovec, Jun 12 2025

cp's OEIS Frontend

A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.

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

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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.

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A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

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A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.

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A352516 Number of excedances (parts above the diagonal) of the n-th composition in standard order.

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A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

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A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

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A352831 Numbers whose weakly increasing prime indices y have exactly one fixed point y(i) = i.