A351012 -id:A351012 - cp's OEIS frontend

A351006 Number of integer partitions of n into parts that are alternately unequal and equal.

1, 1, 1, 2, 3, 4, 4, 6, 7, 9, 10, 12, 14, 16, 18, 22, 25, 28, 31, 36, 40, 46, 50, 56, 64, 71, 78, 88, 96, 106, 118, 130, 143, 158, 172, 190, 209, 228, 248, 274, 298, 324, 354, 384, 418, 458, 494, 536, 584, 631, 683, 742, 800, 864, 936, 1010, 1088, 1176, 1264

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

			The a(1) = 1 through a(11) = 12 partitions (A = 10, B = 11):
  1  2  3   4    5    6    7    8     9      A      B
        21  31   32   42   43   53    54     64     65
            211  41   51   52   62    63     73     74
                 311  411  61   71    72     82     83
                           322  422   81     91     92
                           511  611   522    433    A1
                                3221  711    622    533
                                      4221   811    722
                                      32211  5221   911
                                             42211  4331
                                                    6221
                                                    52211

Without equalities we have A122129, opposite A122135, even-length A351008.
The non-strict version is A351003, opposite A351004, even-length A351012.
The alternately equal and unequal version is A351005, even-length A035457.
The even-length case is A351007.
Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A344605, A350842, A350844, A351011.

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

A122135 Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 26, 31, 40, 48, 60, 72, 89, 106, 130, 154, 186, 220, 264, 310, 370, 433, 512, 598, 704, 818, 958, 1110, 1293, 1494, 1734, 1996, 2308, 2650, 3052, 3496, 4014, 4584, 5248, 5980, 6825, 7760, 8834, 10020, 11380, 12882, 14594

Offset: 0

Michael Somos, Aug 21 2006

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
From Gus Wiseman, Feb 26 2022: (Start)
Conjecture: Also the number of integer partitions y of n such that y_i > y_{i+1} for all even i. For example, the a(1) = 1 through a(9) = 12 partitions are:
  (1)  (2)   (3)   (4)   (5)    (6)     (7)     (8)     (9)
       (11)  (21)  (22)  (32)   (33)    (43)    (44)    (54)
                   (31)  (41)   (42)    (52)    (53)    (63)
                         (221)  (51)    (61)    (62)    (72)
                                (321)   (331)   (71)    (81)
                                (2211)  (421)   (332)   (432)
                                        (3211)  (431)   (441)
                                                (521)   (531)
                                                (3311)  (621)
                                                (4211)  (3321)
                                                        (4311)
                                                        (5211)
The even-length case appears to be A122134.
The odd-length case is A351595.
The alternately unequal version appears to be A122129, even A351008, odd A122130.
The alternately equal version is A351003, even A351012, odd A000009.
The alternately equal and unequal version is A351005, even A035457, odd A351593.
The alternately unequal and equal version is A351006, even A351007, odd A053251. (End)
For Wiseman's conjecture above and three other partition-theoretic interpretations of this sequence see Connor, Proposition 4. - Peter Bala, Jan 02 2025

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
G.f. = q^9 + q^49 + 2*q^89 + 2*q^129 + 3*q^169 + 4*q^209 + 6*q^249 + ...

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.5). MR0858826 (88b:11063)
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(d), p. 591.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Willard G. Connor, Partition Theorems Related to Some Identities of Rogers and Watson, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33-37. MR0541341 (80j:05010). See Theorem 2. [From _N. J. A. Sloane_, Mar 19 2012]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A003242, A035363, A053251, A122129, A122130, A122134.

f:=n->1/mul(1-q^(20*k+n),k=0..20);
f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19);
series(%,q,200); seriestolist(%); # N. J. A. Sloane, Mar 19 2012

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^5] QPochhammer[ x^4, -x^5] QPochhammer[-x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 12 2016 *)
nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+1))*(1 - x^(20*k+2))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+8))*(1 - x^(20*k+9))*(1 - x^(20*k+11))*(1 - x^(20*k+12))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+18))*(1 - x^(20*k+19)) ), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2016 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1) - 1) \2, x^(k^2 + k) / prod(i=1, 2*k+1, 1 - x^i, 1 + x * O(x^(n-k^2-k)))), n))};

Expansion of f(x^2, x^8) / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 12 2016
Expansion of f(-x^3, -x^7) * f(-x^4, -x^16) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [ 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1))).
Let f(n) = 1/Product_{k >= 0} (1-q^(20k+n)). Then g.f. is f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19); - N. J. A. Sloane, Mar 19 2012.
a(n) ~ (3 + sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016

A122134 Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 18, 24, 28, 36, 42, 54, 62, 78, 91, 112, 130, 159, 184, 222, 258, 308, 356, 424, 488, 576, 664, 778, 894, 1044, 1196, 1389, 1590, 1838, 2098, 2419, 2754, 3162, 3596, 4114, 4668, 5328, 6032, 6864, 7760, 8806, 9936, 11252

Offset: 0

Michael Somos, Aug 21 2006, Oct 10 2007

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
In Watson 1937 page 275 he writes "Psi_0(q^{1/2},q) = prod_1^oo (1+q^{2n}) G(-q^2)" so this is the expansion in powers of q^2. - Michael Somos, Jun 29 2015
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
From Gus Wiseman, Feb 26 2022: (Start)
Conjecture: Also the number of even-length integer partitions y of n such that y_i != y_{i+1} for all even i. For example, the a(2) = 1 through a(9) = 7 partitions are:
  (11)  (21)  (22)  (32)  (33)    (43)    (44)    (54)
              (31)  (41)  (42)    (52)    (53)    (63)
                          (51)    (61)    (62)    (72)
                          (2211)  (3211)  (71)    (81)
                                          (3311)  (3321)
                                          (4211)  (4311)
                                                  (5211)
This appears to be the even-length version of A122135.
The odd-length version is A351595.
Cf. A035363, A035457, A350842, A350844, A351003-A351012, A351593. (End)
For Wiseman's conjecture above and three other partition-theoretic interpretations of this sequence see Connor, Proposition 3. - Peter Bala, Jan 02 2025

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = q + q^81 + q^121 + 2*q^161 + 2*q^201 + 4*q^241 + 4*q^281 + ...

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(c), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.6). MR0858826 (88b:11063)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Willard G. Connor, Partition Theorems Related to Some Identities of Rogers and Watson, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
Mircea Merca, From a Rogers's identity to overpartitions, Periodica Mathematica Hungarica, Vol. 75, issue 2, 172-179, 2017.
Michael Somos, Introduction to Ramanujan theta functions
G. N. Watson, The Mock Theta Functions (2), Proceedings of the London Mathematical Society, s2-42: 274-304, 1937.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A053251, A122129, A122130, A122135.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / QPochhammer[ x, x, 2 k], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jun 29 2015 *)
a[ n_] := SeriesCoefficient [ 1 / (QPochhammer[ x^4, -x^5] QPochhammer[ -x, -x^5] QPochhammer[ x, x^2]), {x, 0, n}]; (* Michael Somos, Jun 29 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, -x^5] QPochhammer[ -x^3, -x^5] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Jun 29 2015 *)
nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+2))*(1 - x^(20*k+3))*(1 - x^(20*k+4))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+7))*(1 - x^(20*k+13))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+16))*(1 - x^(20*k+17)) *(1 - x^(20*k+18))), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2016 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n + 1) - 1)\2, x^(k^2 + k) / prod(i=1, 2*k, 1 - x^i, 1 + x * O(x^(n -k^2-k)))), n))};

Euler transform of period 20 sequence [ 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, ...].
Expansion of f(x^4, x^6) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 29 2015
Expansion of f(-x^2, x^3) / phi(-x^2) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Jun 29 2015
Expansion of G(-x) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jun 29 2015
G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k))).
Expansion of f(-x, -x^9) * f(-x^8, -x^12) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is the Ramanujan general theta function.
a(n) = number of partitions of n into parts that are each either == 2, 3, ..., 7 (mod 20) or == 13, 14, ..., 18 (mod 20). - Michael Somos, Jun 29 2015 [corrected by Vaclav Kotesovec, Nov 12 2016]
a(n) ~ (3 - sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016

A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 28, 36, 42, 51, 62, 75, 88, 106, 124, 147, 173, 202, 236, 278, 320, 371, 431, 497, 572, 661, 756, 867, 993, 1132, 1291, 1474, 1672, 1898, 2155, 2439, 2756, 3117, 3512, 3957, 4458, 5008, 5624, 6316, 7072, 7919, 8862, 9899

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The ordered version (compositions) is A027383.
The version for unequal instead of equal is A122135, even-length A351008.
For odd instead of even indices we have A351004, even-length A035363.
Requiring inequalities at odd positions gives A351006, even-length A351007.
The even-length case is A351012.
Cf. A000070, A018819, A088218, A101417, A122129, A350837, A351005.

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,10}]

A351007 Number of even-length integer partitions of n into parts that are alternately unequal and equal.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 10, 13, 14, 16, 18, 20, 23, 27, 28, 32, 37, 40, 44, 51, 54, 60, 67, 73, 81, 90, 96, 107, 118, 127, 139, 154, 166, 181, 198, 213, 232, 256, 273, 297, 325, 348, 377, 411, 440, 476, 516, 555, 598, 647, 692, 746, 807

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.

			The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):
  21  31  32  42  43  53    54    64    65    75    76    86    87
          41  51  52  62    63    73    74    84    85    95    96
                  61  71    72    82    83    93    94    A4    A5
                      3221  81    91    92    A2    A3    B3    B4
                            4221  5221  A1    B1    B2    C2    C3
                                        4331  4332  C1    D1    D2
                                        6221  5331  5332  5441  E1
                                              7221  6331  6332  5442
                                                    8221  7331  6441
                                                          9221  7332
                                                                8331
                                                                A221
                                                                433221

The alternately equal and unequal version is A035457, any length A351005.
This is the even-length case of A351006, odd-length A053251.
Without equalities we have A351008, any length A122129, opposite A122135.
Without inequalities we have A351012, any length A351003, opposite A351004.
Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A350842, A350844, A351011.

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

A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.

			The a(1) = 1 through a(9) = 7 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    221    33      331      44        333
              1111  11111  222     22111    332       441
                           2211    1111111  2222      22221
                           111111           3311      33111
                                            221111    2211111
                                            11111111  111111111

The ordered version (compositions) is A016116.
The even-length case is A035363.
A reverse version is A096441, both A349060.
The version for unequal instead of equal is A122129, even-length A351008.
The version for even instead of odd indices is A351003, even-length A351012.
The strict version is A351005, opposite A351006, even-length A035457.
Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.

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

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

Original entry on oeis.org

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

A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

Original entry on oeis.org

0, 3, 10, 15, 36, 43, 58, 63, 136, 147, 170, 175, 228, 235, 250, 255, 528, 547, 586, 591, 676, 683, 698, 703, 904, 915, 938, 943, 996, 1003, 1018, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2362, 2367, 2696, 2707, 2730, 2735, 2788, 2795, 2810, 2815, 3600, 3619

Offset: 1

Gus Wiseman, Feb 01 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.

			The terms together with their binary expansions and the corresponding compositions begin:
    0:         0  ()
    3:        11  (1,1)
   10:      1010  (2,2)
   15:      1111  (1,1,1,1)
   36:    100100  (3,3)
   43:    101011  (2,2,1,1)
   58:    111010  (1,1,2,2)
   63:    111111  (1,1,1,1,1,1)
  136:  10001000  (4,4)
  147:  10010011  (3,3,1,1)
  170:  10101010  (2,2,2,2)
  175:  10101111  (2,2,1,1,1,1)
  228:  11100100  (1,1,3,3)
  235:  11101011  (1,1,2,2,1,1)
  250:  11111010  (1,1,1,1,2,2)
  255:  11111111  (1,1,1,1,1,1,1,1)

The case of twins (binary weight 2) is A000120.
The Heinz numbers of these compositions are given by A000290.
All terms are evil numbers A001969.
Partitions of this type are counted by A035363, any length A351004.
These compositions are counted by A077957(n-2), see also A016116.
The strict case (distinct twins) is A351009, counted by A032020 with 0's.
The anti-run case is A351011, counted by A003242 interspersed with 0's.
A011782 counts integer compositions.
A085207/A085208 represent concatenation of standard compositions.
A333489 ranks anti-runs, complement A348612.
A345167/A350355/A350356 rank alternating compositions.
A351014 counts distinct runs in standard compositions.
Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],And@@EvenQ/@Length/@Split[stc[#]]&]

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}]

cp's OEIS Frontend

A351006 Number of integer partitions of n into parts that are alternately unequal and equal.

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A122135 Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.

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A122134 Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).

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A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

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A351007 Number of even-length integer partitions of n into parts that are alternately unequal and equal.

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A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

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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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A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

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