A064428 -id:A064428 - cp's OEIS frontend

A342528 Number of compositions with alternating parts weakly decreasing (or weakly increasing).

1, 1, 2, 4, 7, 12, 20, 32, 51, 79, 121, 182, 272, 399, 582, 839, 1200, 1700, 2394, 3342, 4640, 6397, 8771, 11955, 16217, 21878, 29386, 39285, 52301, 69334, 91570, 120465, 157929, 206313, 268644, 348674, 451185, 582074, 748830, 960676, 1229208, 1568716, 1997064

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 strict case (alternating parts are strictly decreasing) is A000041. Is there a bijective proof?
Yes. Construct a Ferrers diagram by placing odd parts horizontally and even parts vertically in a fishbone pattern. The resulting Ferrers diagram will be for an ordinary partition and the process is reversible. It does not appear that this method can be applied to give a formula for this sequence. - Andrew Howroyd, Mar 25 2021

			The a(1) = 1 through a(6) = 20 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (211)   (131)    (51)
                    (1111)  (212)    (141)
                            (221)    (222)
                            (311)    (231)
                            (1211)   (312)
                            (2111)   (321)
                            (11111)  (411)
                                     (1212)
                                     (1311)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (12111)
                                     (21111)
                                     (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Andrew Howroyd)
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The even-length case is A114921.
The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts equal is A342527.
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.
A069916/A342492 = decreasing/increasing first quotients.
A070211/A325546 = weakly decreasing/increasing differences.
A175342/A325545 = constant/distinct differences.
A342495 = constant first quotients (unordered: A342496, strict: A342515, ranking: A342522).
Cf. A001522, A008965, A048004, A059966, A062968, A064410, A064428, A065608, A167606, A325557, A342519.

b:= proc(n, i, j) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, j)+b(n-i, min(n-i, j), min(n-i, i))))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 16 2025

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

seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(1+sum(k=1, n, polcoef(p,k,y)*(polcoef(p,k-1,y) + polcoef(p,k,y))))} \\ Andrew Howroyd, Mar 24 2021

G.f.: Sum_{k>=0} ([y^k] P(x,y))*([y^k] (1 + y)*P(x,y)), where P(x,y) = Product_{k>=1} 1/(1 - y*x^k). - Andrew Howroyd, Jan 16 2025

Terms a(21) and beyond from Andrew Howroyd, Mar 24 2021

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&]

A342192 Heinz numbers of partitions of crank 0.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 100, 106, 118, 122, 134, 140, 142, 146, 158, 166, 178, 194, 196, 202, 206, 214, 218, 220, 226, 254, 260, 262, 274, 278, 298, 300, 302, 308, 314, 326, 334, 340, 346, 358, 362, 364, 380, 382, 386, 394, 398

Offset: 1

Gus Wiseman, Apr 05 2021

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.

See A257989 or the program for a definition of crank of a partition.

			The sequence of terms together with their prime indices begins:
      6: {1,2}        106: {1,16}       218: {1,29}
     10: {1,3}        118: {1,17}       220: {1,1,3,5}
     14: {1,4}        122: {1,18}       226: {1,30}
     22: {1,5}        134: {1,19}       254: {1,31}
     26: {1,6}        140: {1,1,3,4}    260: {1,1,3,6}
     34: {1,7}        142: {1,20}       262: {1,32}
     38: {1,8}        146: {1,21}       274: {1,33}
     46: {1,9}        158: {1,22}       278: {1,34}
     58: {1,10}       166: {1,23}       298: {1,35}
     62: {1,11}       178: {1,24}       300: {1,1,2,3,3}
     74: {1,12}       194: {1,25}       302: {1,36}
     82: {1,13}       196: {1,1,4,4}    308: {1,1,4,5}
     86: {1,14}       202: {1,26}       314: {1,37}
     94: {1,15}       206: {1,27}       326: {1,38}
    100: {1,1,3,3}    214: {1,28}       334: {1,39}

Indices of zeros in A257989.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A001522 counts partitions of positive crank.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
Cf. A000726, A008965, A056239, A112798, A124010, A130091, A325351, A325352.

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

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&]

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

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}]

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

cp's OEIS Frontend

A342528 Number of compositions with alternating parts weakly decreasing (or weakly increasing).

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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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A342192 Heinz numbers of partitions of crank 0.

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

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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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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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