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.

%I A351008 #24 Jan 04 2025 10:37:49
%S A351008 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,
%T A351008 142,168,197,231,271,316,369,429,497,577,668,770,888,1023,1175,1348,
%U A351008 1545,1767,2020,2306,2626,2990,3401,3860,4379,4963,5616,6350,7173,8093
%N A351008 Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.
%C A351008 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
%F A351008 Conjecture: a(n+1) = A122129(n+1) - A122130(n). - _Gus Wiseman_, Feb 21 2022
%F A351008 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
%e A351008 The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):
%e A351008   21   31   32   42   43   53     54     64     65     75     76
%e A351008             41   51   52   62     63     73     74     84     85
%e A351008                       61   71     72     82     83     93     94
%e A351008                            3221   81     91     92     A2     A3
%e A351008                                   4221   4321   A1     B1     B2
%e A351008                                          5221   4331   4332   C1
%e A351008                                                 5321   5331   5332
%e A351008                                                 6221   5421   5431
%e A351008                                                        6321   6331
%e A351008                                                        7221   6421
%e A351008                                                               7321
%e A351008                                                               8221
%e A351008 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
%p A351008 series(add(q^(n*(n+2))/mul(1 - q^k, k = 1..2*n), n = 0..10), q, 141):
%p A351008 seq(coeftayl(%, q = 0, n), n = 0..140); # _Peter Bala_, Jan 03 2025
%t A351008 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]
%Y A351008 The version for equal instead of unequal is A035363.
%Y A351008 The alternately equal and unequal version is A035457, any length A351005.
%Y A351008 This is the even-length case of A122129, opposite A122135.
%Y A351008 The odd-length version appears to be A122130.
%Y A351008 The alternately unequal and equal version is A351007, any length A351006.
%Y A351008 Cf. A000070, A003242, A018819, A027383, A053251, A122134, A350842, A350844, A351003, A351004, A351012.
%K A351008 nonn,easy
%O A351008 0,6
%A A351008 _Gus Wiseman_, Jan 31 2022