%I #12 Apr 17 2021 01:53:20

%S 1,1,1,3,3,5,8,10,13,18,27,32,44,55,73,97,121,151,194,240,299,384,465,

%T 576,706,869,1051,1293,1572,1896,2290,2761,3302,3973,4732,5645,6759,

%U 7995,9477,11218,13258,15597,18393,21565,25319,29703,34701,40478,47278,54985

%N Number of strict compositions of n with alternating parts strictly decreasing.

%C These are finite odd-length sequences q of distinct positive integers summing to n such that q(i) > q(i+2) for all possible i.

%H Andrew Howroyd, <a href="/A342343/b342343.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{k>=0} binomial(k,floor(k/2)) * [y^k](Product_{j>=1} 1 + y*x^j). - _Andrew Howroyd_, Apr 16 2021

%e The a(1) = 1 through a(8) = 13 compositions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (1,2) (1,3) (1,4) (1,5) (1,6) (1,7)

%e (2,1) (3,1) (2,3) (2,4) (2,5) (2,6)

%e (3,2) (4,2) (3,4) (3,5)

%e (4,1) (5,1) (4,3) (5,3)

%e (2,3,1) (5,2) (6,2)

%e (3,1,2) (6,1) (7,1)

%e (3,2,1) (2,4,1) (2,5,1)

%e (4,1,2) (3,4,1)

%e (4,2,1) (4,1,3)

%e (4,3,1)

%e (5,1,2)

%e (5,2,1)

%t ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];

%t Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],ici]],{n,0,15}]

%o (PARI) seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=0, n, binomial(k, k\2) * polcoef(p,k,y)))} \\ _Andrew Howroyd_, Apr 16 2021

%Y The non-strict case is A000041 (see A342528 for a bijective proof).

%Y The non-strict odd-length case is A001522.

%Y Strict compositions in general are counted by A032020

%Y The non-strict even-length case is A064428.

%Y The case of reversed partitions is A065033.

%Y A000726 counts partitions with alternating parts unequal.

%Y A003242 counts anti-run compositions.

%Y A027193 counts odd-length compositions.

%Y A034008 counts even-length compositions.

%Y A064391 counts partitions by crank.

%Y A064410 counts partitions of crank 0.

%Y A224958 counts compositions with alternating parts unequal.

%Y A257989 gives the crank of the partition with Heinz number n.

%Y A325548 counts compositions with strictly decreasing differences.

%Y A342194 counts strict compositions with equal differences.

%Y A342527 counts compositions with alternating parts equal.

%Y Cf. A000009, A000670, A008965, A062968, A065608, A109613, A114921, A325546, A332304, A332305, A342532.

%K nonn

%O 0,4

%A _Gus Wiseman_, Apr 01 2021