A064410 -id:A064410 - cp's OEIS frontend

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

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&]

A342527 Number of compositions of n with alternating parts equal.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 11, 12, 16, 17, 21, 20, 29, 24, 31, 32, 38, 32, 46, 36, 51, 46, 51, 44, 69, 51, 61, 60, 73, 56, 87, 60, 84, 74, 81, 76, 110, 72, 91, 88, 115, 80, 123, 84, 117, 112, 111, 92, 153, 101, 132, 116, 139, 104, 159, 120, 161, 130, 141, 116, 205, 120, 151, 156, 178, 142, 195, 132, 183, 158

Offset: 0

Gus Wiseman, Mar 24 2021

nonn

These are finite sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (121)   (41)     (42)      (43)       (44)
                    (1111)  (131)    (51)      (52)       (53)
                            (212)    (141)     (61)       (62)
                            (11111)  (222)     (151)      (71)
                                     (1212)    (232)      (161)
                                     (2121)    (313)      (242)
                                     (111111)  (12121)    (323)
                                               (1111111)  (1313)
                                                          (2222)
                                                          (3131)
                                                          (21212)
                                                          (11111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The odd-length case is A062968.
The even-length case is A065608.
The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts weakly decreasing is A342528.
A000005 counts constant compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A175342 counts compositions with constant differences.
A342495 counts compositions with constant first quotients.
A342496 counts partitions with constant first quotients (strict: A342515, ranking: A342522).
Cf. A001522, A008965, A048004, A059966, A064410, A064428, A069916, A114921, A167606, A325545, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

a(n) = 1 + n + A000203(n) - 2*A000005(n).

a(n) = A065608(n) + A062968(n).

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

A114921 Number of unimodal compositions of n+2 where the maximal part appears exactly twice.

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 11, 16, 27, 40, 63, 92, 141, 202, 299, 426, 614, 862, 1222, 1694, 2362, 3242, 4456, 6054, 8229, 11072, 14891, 19872, 26477, 35050, 46320, 60866, 79827, 104194, 135703, 176008, 227791, 293702, 377874, 484554, 620011, 790952, 1006924

Offset: 0

Michael Somos, Jan 07 2006

nonn

Old name was: Expansion of a q-series.
a(n) is also the number of 2-colored partitions of n with the same number of parts in each color. - Shishuo Fu, May 30 2017
From Gus Wiseman, Mar 25 2021: (Start)
Also the number of even-length compositions of n with alternating parts weakly decreasing. Allowing odd lengths also gives A342528. The version with alternating parts strictly decreasing appears to be A064428. The a(2) = 1 through a(7) = 16 compositions are:
  (1,1)  (1,2)  (1,3)      (1,4)      (1,5)          (1,6)
         (2,1)  (2,2)      (2,3)      (2,4)          (2,5)
                (3,1)      (3,2)      (3,3)          (3,4)
                (1,1,1,1)  (4,1)      (4,2)          (4,3)
                           (1,2,1,1)  (5,1)          (5,2)
                           (2,1,1,1)  (1,2,1,2)      (6,1)
                                      (1,3,1,1)      (1,3,1,2)
                                      (2,1,2,1)      (1,4,1,1)
                                      (2,2,1,1)      (2,2,1,2)
                                      (3,1,1,1)      (2,2,2,1)
                                      (1,1,1,1,1,1)  (2,3,1,1)
                                                     (3,1,2,1)
                                                     (3,2,1,1)
                                                     (4,1,1,1)
                                                     (1,2,1,1,1,1)
                                                     (2,1,1,1,1,1)
(End)

			From _Joerg Arndt_, Jun 10 2013: (Start)
There are a(7)=16 such compositions of 7+2=9 where the maximal part appears twice:
  01:  [ 1 1 1 1 1 2 2 ]
  02:  [ 1 1 1 1 2 2 1 ]
  03:  [ 1 1 1 2 2 1 1 ]
  04:  [ 1 1 1 3 3 ]
  05:  [ 1 1 2 2 1 1 1 ]
  06:  [ 1 1 3 3 1 ]
  07:  [ 1 2 2 1 1 1 1 ]
  08:  [ 1 2 3 3 ]
  09:  [ 1 3 3 1 1 ]
  10:  [ 1 3 3 2 ]
  11:  [ 1 4 4 ]
  12:  [ 2 2 1 1 1 1 1 ]
  13:  [ 2 3 3 1 ]
  14:  [ 3 3 1 1 1 ]
  15:  [ 3 3 2 1 ]
  16:  [ 4 4 1 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
S. Fu and D. Tang, On a generalized crank for k-colored partitions, arXiv:1705.10067 [math.CO], 2017.
B. Kim and J. Lovejoy, Ramanujan-type partial theta identities and rank differences for special unimodal sequences, Annals of Combinatorics, 19 (2015), 705-733.

Cf. A226541 (max part appears three times), A188674 (max part m appears m times), A001523 (max part appears any number of times).
Column k=2 of A247255.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A034008 counts even-length compositions.
A065608 counts even-length compositions with alternating parts equal.
A342528 counts compositions with alternating parts weakly decreasing.
A342532 counts even-length compositions with alternating parts unequal.
Cf. A000726, A001522, A008965, A062968, A064410, A064428, A069916, A070211, A175342, A224958, A342495, A342527.

max = 50; s = (1+Sum[2*(-1)^k*q^(k(k+1)/2), {k, 1, max}])/QPochhammer[q]^2+ O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, from 1st g.f. *)
wdw[q_]:=And@@Table[q[[i]]>=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],wdw]],{n,0,15}] (* Gus Wiseman, Mar 25 2021 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, n\2, x^(2*k) / prod(i=1, k, 1 - x^i, 1 + x * O(x^n))^2), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, sqrtint(8*n + 1)\2, 2*(-1)^k * x^((k^2+k)/2), 1 + A) / eta(x + A)^2, n))};

G.f.: 1 + Sum_{k>0} (x^k / ((1-x)(1-x^2)...(1-x^k)))^2 = (1 + Sum_{k>0} 2 (-1)^k x^((k^2+k)/2) ) / (Product_{k>0} (1 - x^k))^2.
G.f.: 1 + x*(1 - G(0))/(1-x) where G(k) = 1 - x/(1-x^(k+1))^2/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A006330(n) - A001523(n). - Vaclav Kotesovec, Jun 22 2015
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (16 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Oct 24 2018

New name from Joerg Arndt, Jun 10 2013

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

A343347 Number of strict integer partitions of n with a part divisible by all the others.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 5, 4, 6, 6, 6, 8, 7, 7, 10, 9, 9, 12, 10, 8, 11, 11, 10, 14, 13, 11, 13, 12, 15, 20, 17, 15, 19, 19, 19, 22, 18, 17, 23, 22, 22, 28, 25, 24, 31, 28, 26, 32, 32, 30, 34, 32, 29, 37, 33, 27, 36, 33, 34, 44, 38, 36, 45, 45

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   632  93    841  D1   C3
                                 621  631  821  A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

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

The dual version is A097986 (non-strict: A083710).
The non-strict version is A130689 (Heinz numbers: complement of A343337).
The strict complement is counted by A343377.
The case with smallest part divisible by all the others is A343378.
The case with smallest part not divisible by all the others is A343380.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021

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

A352832 Number of reversed integer partitions y of n with exactly one fixed point y(i) = i.

Original entry on oeis.org

0, 1, 1, 1, 4, 3, 7, 7, 14, 19, 24, 32, 46, 60, 85, 109, 140, 179, 239, 300, 397, 495, 636, 790, 995, 1239, 1547, 1926, 2396, 2942, 3643, 4432, 5435, 6602, 8038, 9752, 11842, 14292, 17261, 20714, 24884, 29733, 35576, 42375, 50522, 60061, 71363, 84551, 100101

Offset: 0

Gus Wiseman, Apr 08 2022

nonn

A reversed integer partition of n is a finite weakly increasing sequence of positive integers summing to n.

			The a(0) = 0 through a(8) = 14 partitions (empty column indicated by dot):
  .  (1)  (11)  (111)  (13)    (14)     (15)      (16)       (17)
                       (22)    (1112)   (114)     (115)      (116)
                       (112)   (11111)  (222)     (1123)     (134)
                       (1111)           (1113)    (11113)    (224)
                                        (1122)    (11122)    (233)
                                        (11112)   (111112)   (1115)
                                        (111111)  (1111111)  (2222)
                                                             (11114)
                                                             (11123)
                                                             (11222)
                                                             (111113)
                                                             (111122)
                                                             (1111112)
                                                             (11111111)
For example, the reversed partition (2,2,4) has a unique fixed point at the second position.

* = unproved
*The non-reverse version is A001522, ranked by A352827, strict A352829.
*The non-reverse complement is A064428, ranked by A352826, strict A352828.
This is column k = 1 of A238352.
For no fixed point: counted by A238394, ranked by A352830, strict A025147.
For > 0 fixed points: counted by A238395, ranked by A352872, strict A096765.
The version for compositions is A240736, complement A352520.
These partitions are ranked by A352831.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, nonfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A352822 counts fixed points of prime indices.
A352833 counts partitions by fixed points.
Cf. A064410, A177510, A257990, A325187, A351983, A352824, A352825.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Reverse/@IntegerPartitions[n],pq[#]==1&]],{n,0,30}]

A064391 Triangle T(n,k) with zeroth row {1} and row n for n >= 1 giving number of partitions of n with crank k, for -n <= k <= n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 29 2001

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

n-th row contains 2n+1 terms.

			{T(20, k), -20 <= k <=20} = {1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 19, 20, 26, 28, 34, 34, 39, 38, 41, 38, 39, 34, 34, 28, 26, 20, 19, 13, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1}.
From _Omar E. Pol_, Mar 04 2012: (Start)
Triangle begins:
.                          1;
.                       1, 0, 0;
.                    1, 0, 0, 0, 1;
.                 1, 0, 0, 1, 0, 0, 1;
.              1, 0, 1, 0, 1, 0, 1, 0, 1;
.           1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1;
.        1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1;
.     1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1;
.  1, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1;
1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 3, 2, 3, 1, 2, 1, 1, 0, 1;
(End)

G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
F. Garvan, New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc., 305 (1988), 47-77.

Cf. A001522, A064410, A064428.

Row sums give A000041. - Omar E. Pol, Mar 04 2012

max = 12; f[k_ /; k < 0] := f[-k]; f[k_] := Sum[(-1)^m*x^(k*m)*(x^((m^2 + m)/2) - x^((m^2 - m)/2)), {m, 1, max}]/Product[1 - x^m, {m, 1, max}]; t = Table[ Series[f[k], {x, 0, max}] // CoefficientList[#, x]&, {k, -(max-2), max-2}] // Transpose; Table[If[n == 2, {1, 0, 0}, Table[t[[n, k]], {k, max-n, max+n-2}]], {n, 1, max-1}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Vladeta Jovovic *)

for n in (0..9): # computes the sequence as a triangle
    a = [p.crank() for p in Partitions(n)]
    [a.count(k) for k in (-n..n)] # Peter Luschny, Sep 15 2014

G.f. for k-th column is Sum(m>=1, (-1)^m*x^(k*m)*(x^((m^2+m)/2)-x^((m^2-m)/2)))/Product(m>=1, 1-x^m). - Vladeta Jovovic, Dec 22 2004

More terms from Vladeta Jovovic, Sep 29 2001

cp's OEIS Frontend

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

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A342527 Number of compositions of n with alternating parts equal.

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

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A114921 Number of unimodal compositions of n+2 where the maximal part appears exactly twice.

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

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A343347 Number of strict integer partitions of n with a part divisible by all the others.

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

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

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