A332010 - OEIS

%I #17 Mar 30 2024 23:08:33

%S 1,2,4,1,1,8,3,4,16,104,40,24,1,32,392,896,480,54,64,152,308,1496,63,

%T 108,1,128,2276,30384,14410,4315,3024,14,256,7340,153400,235252,24766,

%U 180416,4984,448,1,512,23172,365520,1713160,5794944,3739512,881152,63840,453

%N Irregular triangle of denominators of the average value of the first letter over all derangements of {1, 2, ..., n} with k descents.

%C Even-indexed rows have length n - 1; odd-indexed rows have length n - 2.

%C Conjecture: T(n, 1) = 2^(n-2).

%C T(2n, 2n-1) = 1, since there is only one derangement of 2n letters with 2n-1 descents.

%C The analogous sequence for permutations is T'(n, k) = 1. That is, the expected value of the first letter of a permutation with k descents is an integer (namely k + 1).

%e Triangle begins:

%e     1;

%e     2;

%e     4,    1,     1;

%e     8,    3,     4;

%e    16,  104,    40,    24,    1;

%e    32,  392,   896,   480,   54;

%e    64,  152,   308,  1496,   63,  108,  1;

%e   128, 2276, 30384, 14410, 4315, 3024, 14.

%e T(4,1) = 4 because the derangements of four letters with one descent are

%e [2,3,4,1], [2,4,1,3], [3,4,1,2], and [4,1,2,3], and the expected value of the first letter is (2+2+3+4)/4 = 11/4, which has 4 as its denominator.

%Y Cf. A000166, A083329, A219836.

%Y Numerators are given by A332009.

%K nonn,tabf

%O 2,2

%A _Peter Kagey_, Feb 04 2020