A351006 -id:A351006 - cp's OEIS frontend

A351008 Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 23, 28, 34, 41, 50, 60, 71, 85, 102, 120, 142, 168, 197, 231, 271, 316, 369, 429, 497, 577, 668, 770, 888, 1023, 1175, 1348, 1545, 1767, 2020, 2306, 2626, 2990, 3401, 3860, 4379, 4963, 5616, 6350, 7173, 8093

Offset: 0

Gus Wiseman, Jan 31 2022

Write the series in the g.f. given below as Sum_{k >= 0} q^(1 + 3 + 5 + ... + 2*k-1 + 2*k)/Product_{i = 1..2*k} 1 - q^i. Since 1/Product_{i = 1..2*k} 1 - q^i is the g.f. for partitions with parts <= 2*k, we see that the k-th summand of the series is the g.f. for partitions with largest part 2*k in which every odd number less than 2*k appears at least once as a part. The partitions of this type are conjugate to (and hence equinumerous with) the partitions (y_1, y_2, ..., y_{2*k}) of even length 2*k having strict decrease y_i > y_(i+1) for all odd i < 2*k. - Peter Bala, Jan 02 2024

			The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):
  21   31   32   42   43   53     54     64     65     75     76
            41   51   52   62     63     73     74     84     85
                      61   71     72     82     83     93     94
                           3221   81     91     92     A2     A3
                                  4221   4321   A1     B1     B2
                                         5221   4331   4332   C1
                                                5321   5331   5332
                                                6221   5421   5431
                                                       6321   6331
                                                       7221   6421
                                                              7321
                                                              8221
a(10) = 6: the six partitions 64, 73, 82, 91, 4321 and 5221 listed above have conjugate partitions 222211, 2221111, 22111111, 211111111, 4321 and 43111, These are the partitions of 10 with largest part L even and such that every odd number less than L appears at least once as a part. - _Peter Bala_, Jan 02 2024

The version for equal instead of unequal is A035363.
The alternately equal and unequal version is A035457, any length A351005.
This is the even-length case of A122129, opposite A122135.
The odd-length version appears to be A122130.
The alternately unequal and equal version is A351007, any length A351006.
Cf. A000070, A003242, A018819, A027383, A053251, A122134, A350842, A350844, A351003, A351004, A351012.

series(add(q^(n*(n+2))/mul(1 - q^k, k = 1..2*n), n = 0..10), q, 141):
seq(coeftayl(%, q = 0, n), n = 0..140); # Peter Bala, Jan 03 2025

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

Conjecture: a(n+1) = A122129(n+1) - A122130(n). - Gus Wiseman, Feb 21 2022

G.f.: Sum_{n >= 0} q^(n*(n+2))/Product_{k = 1..2*n} 1 - q^k = 1 + q^3 + q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 4*q^8 + 5*q^9 + 6*q^10 + .... - Peter Bala, Jan 02 2024

A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 9, 10, 13, 16, 21, 24, 29, 35, 43, 50, 60, 70, 83, 97, 113, 132, 156, 178, 206, 239, 275, 316, 365, 416, 477, 545, 620, 706, 806, 912, 1034, 1173, 1326, 1496, 1691, 1902, 2141, 2410, 2704, 3034, 3406, 3808, 4261, 4765, 5317, 5932, 6617

Offset: 0

Gus Wiseman, Feb 03 2022

nonn

			The a(2) = 1 through a(8) = 9 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (3111)    (2221)    (71)
                              (111111)  (4111)    (2222)
                                        (211111)  (3221)
                                                  (5111)
                                                  (311111)
                                                  (11111111)

The ordered version (compositions) is A027383(n-2).
For odd instead of even indices we have A035363, any length A351004.
The version for unequal parts appears to be A122134, any length A122135.
This is the even-length case of A351003.
Requiring inequalities at odd positions gives A351007, any length A351006.
Cf. A000070, A018819, A035457, A088218, A101417, A122129, A350837, A351005, A351008.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]

A357635 Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.

Original entry on oeis.org

2, 8, 24, 32, 54, 128, 135, 162, 375, 384, 512, 648, 864, 875, 1250, 1715, 1944, 2048, 2160, 2592, 3773, 4374, 4802, 5000, 6000, 6144, 8192, 9317, 10368, 10935, 13122, 13824, 14000, 15000, 17303, 19208, 20000, 24167, 27440, 29282, 30375, 31104, 32768, 33750

Offset: 1

Gus Wiseman, Oct 28 2022

nonn

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

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:
    2: {1}
    8: {1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   54: {1,2,2,2}
  128: {1,1,1,1,1,1,1}
  135: {2,2,2,3}
  162: {1,2,2,2,2}
  375: {2,3,3,3}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}
  648: {1,1,1,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  875: {3,3,3,4}

The version for k = 0 is A000583, standard compositions A357625-A357626.
The version for original alternating sum is A345958.
Positions of ones in A357633, non-reverse A357629.
The skew version for k = 0 is A357636, non-reverse A357632.
These partitions are counted by A035444, skew A035544.
The non-reverse version is A357851, k = 0 version A357631.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even-length A357642.
Cf. A000290, A003963, A053251, A055932, A357621-A357624, A357630, A357634, A357637, A357639, A357640.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Select[Range[1000],halfats[Reverse[primeMS[#]]]==1&]

A357645 Triangle read by rows where T(n,k) is the number of integer compositions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 2, 2, 4, 0, 0, 3, 5, 3, 5, 0, 0, 4, 8, 10, 4, 6, 0, 0, 5, 11, 18, 18, 5, 7, 0, 0, 6, 14, 28, 36, 30, 6, 8, 0, 0, 7, 17, 41, 63, 65, 47, 7, 9, 0, 0, 8, 20, 58, 104, 126, 108, 70, 8, 10, 0, 0, 9, 23, 80, 164, 230, 230, 168, 100, 9, 11

Offset: 0

Gus Wiseman, Oct 12 2022

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

			Triangle begins:
   1
   0   1
   0   0   2
   0   0   1   3
   0   0   2   2   4
   0   0   3   5   3   5
   0   0   4   8  10   4   6
   0   0   5  11  18  18   5   7
   0   0   6  14  28  36  30   6   8
   0   0   7  17  41  63  65  47   7   9
   0   0   8  20  58 104 126 108  70   8  10
Row n = 6 counts the following compositions:
  (114)   (123)    (132)     (141)  (6)
  (1113)  (213)    (222)     (231)  (15)
  (1122)  (1212)   (312)     (321)  (24)
  (1131)  (1221)   (1311)    (411)  (33)
          (2112)   (2211)           (42)
          (2121)   (3111)           (51)
          (11121)  (11112)
          (11211)  (12111)
                   (21111)
                   (111111)

Row sums are A011782.
For original alternating sum we have A097805, unordered A344651.
Column k = n-4 appears to be A177787.
The case of partitions is A357637, skew A357638.
The central column k=0 is A357641 (aerated).
The skew-alternating version is A357646.
The reverse version for partitions is A357704, skew A357705.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A029862, A035363, A035544, A357136, A357631, A357639.

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],halfats[#]==k&]],{n,0,10},{k,-n,n,2}]

A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Oct 10 2022

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  2  0  3
  0  0  2  2  0  3
  0  0  3  1  3  0  4
  0  0  3  2  4  2  0  4
  0  0  4  2  6  2  3  0  5
  0  0  4  3  5  7  3  3  0  5
  0  0  5  3  8  4 10  2  4  0  6
  0  0  5  4  8  6 11  9  3  4  0  6
  0  0  6  4 11  5 15  8 13  3  5  0  7
  0  0  6  5 11  8 13 19 10 13  4  5  0  7
  0  0  7  5 14  8 19 13 25  9 17  4  6  0  8
  0  0  7  6 14 11 19 17 29 23 13 18  5  6  0  8
Row n = 7 counts the following reversed partitions:
  .  .  (115)   (124)   (133)      (11113)   .  (7)
        (1114)  (1222)  (223)      (111112)     (16)
        (1123)          (11122)                 (25)
                        (1111111)               (34)

Row sums are A000041.
Last entry of row n is A008619(n).
The central column in the non-reverse case is A035363, skew A035544.
For original reverse-alternating sum we have A344612.
For original alternating sum we have A344651, ordered A097805.
The non-reverse version is A357637, skew A357638.
The central column is A357639, skew A357640.
The non-reverse ordered version (compositions) is A357645, skew A357646.
The skew-alternating version is A357705.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n],halfats[#]==k&]],{n,0,15},{k,-n,n,2}]

A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Oct 10 2022

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  3  1  2  0  1
  0  3  2  3  2  0  1
  0  4  2  4  1  3  0  1
  0  4  3  3  6  2  3  0  1
  0  5  3  5  3  7  2  4  0  1
  0  5  4  5  4  9  7  3  4  0  1
  0  6  4  7  3 12  5 10  3  5  0  1
  0  6  5  7  5 10 16  7 11  4  5  0  1
  0  7  5  9  5 14 11 18  7 14  4  6  0  1
Row n = 7 counts the following reversed partitions:
  .  (16)   (25)   (34)       (1123)  (1114)   .  (7)
     (115)  (223)  (1222)             (11113)
     (124)         (111112)           (11122)
     (133)         (1111111)

Row sums are A000041.
First nonzero entry of each row is A004526.
The central column is A357640, half A357639.
For original alternating sum we have A344651, ordered A097805.
The half-alternating version is A357704.
The ordered non-reverse version (compositions) is A357646, half A357645.
The non-reverse version is A357638, half A357637.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A035363, A035594, A053251, A298311, A357136, A357189, A357487, A357488, A357624, A357631, A357632, A357636.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n],skats[#]==k&]],{n,0,11},{k,-n,n,2}]

A239327 Number of palindromic Carlitz compositions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 5, 5, 7, 10, 14, 14, 25, 26, 42, 48, 75, 79, 132, 142, 226, 252, 399, 432, 704, 760, 1223, 1336, 2143, 2328, 3759, 4079, 6564, 7150, 11495, 12496, 20135, 21874, 35215, 38310, 61639, 67018, 107912, 117298, 188839, 205346, 330515, 359350, 578525, 628951

Offset: 0

Geoffrey Critzer, Mar 16 2014

nonn

A palindromic composition is a composition that is identical to its own reverse. There are 2^floor(n/2) palindromic compositions. A Carlitz composition has no two consecutive equal parts (A003242). This sequence enumerates compositions that are both palindromic and Carlitz.

Also the number of odd-length integer compositions of n into parts that are alternately unequal and equal (n > 0). The unordered version (partitions) is A053251. - Gus Wiseman, Feb 26 2022

			a(9) = 7 because we have: 9, 1+7+1, 2+5+2, 4+1+4, 1+3+1+3+1, 2+1+3+1+2, 1+2+3+2+1. 2+3+4 is not counted because it is not palindromic. 3+3+3 is not counted because it has consecutive equal parts.

S. Heubach and T. Mansour, Compositions of n with parts in a set, Congr. Numer. 168 (2004), 127-143.
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2010, page 67.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Petros Hadjicostas, Cyclic, Dihedral and Symmetrical Carlitz Compositions of a Positive Integer, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.5.

Carlitz compositions are counted by A003242.
Palindromic compositions are counted by A016116.
The unimodal case is A096441.
Cf. A053251, A122129, A122130, A351003, A351006, A351007.

b:= proc(n, i) option remember; `if`(i=0, 0, `if`(n=0, 1,
      add(`if`(i=j, 0, b(n-j, j)), j=1..n)))
    end:
a:= n-> `if`(n=0, 1, add(b(i, n-2*i), i=0..n/2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 16 2014

nn=50;CoefficientList[Series[(1+Sum[x^j(1-x^j)/(1+x^(2j)),{j,1,nn}])/(1-Sum[x^(2j)/(1+x^(2j)),{j,1,nn}]),{x,0,nn}],x]
(* or *)
Table[Length[Select[Level[Map[Permutations,Partitions[n]],{2}],Apply[And,Table[#[[i]]==#[[Length[#]-i+1]],{i,1,Floor[Length[#]/2]}]]&&Apply[And,Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1}]]&]],{n,0,20}]

a(n) = polcoeff((1 + sum(j=1, n, x^j*(1-x^j)/(1+x^(2*j)) + O(x*x^n))) / (1 - sum(j=1, n, x^(2*j)/(1+x^(2*j)) + O(x*x^n))), n); \\ Andrew Howroyd, Oct 12 2017

G.f.: (1 + Sum_{j>=1} x^j*(1-x^j)/(1+x^(2*j))) / (1 - Sum_{j>=1} x^(2*j)/(1+x^(2*j))).

a(n) ~ c / r^n, where r = 0.7558768372943356987836792261127971643747976345582722756032673... is the root of the equation sum_{j>=1} x^(2*j)/(1+x^(2*j)) = 1, c = 0.5262391407444644722747255167331403939384758635340487280277... if n is even and c = 0.64032989654153238794063877354074732669441634551692765196197... if n is odd. - Vaclav Kotesovec, Aug 22 2014

A351595 Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 30, 35, 44, 52, 63, 74, 90, 105, 126, 148, 175, 204, 242, 280, 330, 382, 446, 515, 600, 690, 800, 919, 1060, 1214, 1398, 1595, 1830, 2086, 2384, 2711, 3092, 3506, 3988, 4516, 5122, 5788, 6552, 7388, 8345

Offset: 0

Gus Wiseman, Feb 25 2022

nonn

			The a(1) = 1 through a(12) = 10 partitions (A..C = 10..12):
  1   2   3   4   5     6     7     8     9     A     B       C
                  221   321   331   332   432   442   443     543
                              421   431   441   532   542     552
                                    521   531   541   551     642
                                          621   631   632     651
                                                721   641     732
                                                      731     741
                                                      821     831
                                                      33221   921
                                                              43221

The ordered version (compositions) is A000213 shifted right once.
All odd-length partitions are counted by A027193.
The opposite appears to be A122130, even-length A351008, any length A122129.
This appears to be the odd-length case of A122135, even-length A122134.
The case that is constant at odd indices:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
For equality instead of inequality:
- any length: A351003
- odd-length: A000009 (except at 0)
- even-length: A351012
- opposite any length: A351004
- opposite odd-length: A351594
- opposite even-length: A035363
Cf. A000041, A000070, A003242, A027383, A236559, A236914, A350842.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]>#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]

A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318

Offset: 0

Gus Wiseman, Feb 23 2022

nonn

Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.

			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
              221     331  332  441  442  443    552  553    554  663
                                          551         661    662  771
                                          33221       44221       44331
                                                                  55221

The even-length ordered version is A003242, ranked by A351010.
The opposite version is A053251, even-length A351007, any length A351006.
This is the odd-length case of A351005, even-length A035457.
With only equalities we get:
  - opposite any length: A351003
  - opposite odd-length: A000009  (except at 0)
  - opposite even-length: A351012
- any length: A351004
- odd-length: A351594
- even-length: A035363
Without equalities we get:
  - opposite any length: A122129 (apparently)
  - opposite odd-length: A122130 (apparently)
  - opposite even-length: A351008
- any length: A122135 (apparently)
- odd-length: A351595
- even-length: A122134 (apparently)
Cf. A000070, A000984, A027383, A053738, A236559, A236914, A350842, A350844.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[If[EvenQ[i],#[[i]]!=#[[i+1]],#[[i]]==#[[i+1]]],{i,Length[#]-1}]&]],{n,0,30}]

A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 9, 4, 13, 6, 19, 6, 26, 10, 35, 12, 49, 16, 64, 20, 87, 27, 115, 32, 151, 44, 195, 53, 256, 69, 328, 84, 421, 108, 537, 130, 682, 167, 859, 202, 1085, 252, 1354, 305, 1694, 380, 2104, 456, 2609, 564, 3218, 676, 3968, 826, 4863

Offset: 0

Gus Wiseman, Feb 24 2022

nonn

These are partitions with all even run-lengths except for the last, which is odd.

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)    (9)
            (111)       (221)    (222)  (331)      (332)  (333)
                        (11111)         (22111)           (441)
                                        (1111111)         (22221)
                                                          (33111)
                                                          (2211111)
                                                          (111111111)

The ordered version (compositions) is A016116 shifted right once.
All odd-length partitions are counted by A027193.
The opposite version is A117409, even-length A351012, any length A351003.
Replacing equal with unequal relations appears to give:
- any length: A122129
- odd length: A122130
- even length: A351008
- opposite any length: A122135
- opposite odd length: A351595
- opposite even length: A122134
This is the odd-length case of A351004, even-length A035363.
The case that is also strict at even indices is:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
A reverse version is A096441; see also A349060.
Cf. A000009, A000041, A000070, A000984, A003242, A027383, A053738, A236559, A236914, A350842.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

cp's OEIS Frontend

A351008 Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.

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A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

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A357635 Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.

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A357645 Triangle read by rows where T(n,k) is the number of integer compositions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

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A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

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A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

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A239327 Number of palindromic Carlitz compositions of n.

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A351595 Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.

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A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

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