A219836 -id:A219836 - cp's OEIS frontend

A332009 Irregular triangle of numerators of the average value of the first letter over all derangements of {1, 2, ..., n} with k descents.

2, 5, 11, 3, 4, 23, 10, 17, 47, 367, 173, 121, 6, 95, 1431, 3973, 2513, 331, 191, 567, 1395, 8011, 391, 761, 8, 383, 8615, 139897, 78483, 27201, 21685, 113, 767, 28073, 715277, 1298393, 158189, 1310499, 40613, 4049, 10, 1535, 89311, 1721203, 9555617, 37414903, 27455053, 7254087, 582767, 4549

Offset: 2

Peter Kagey, Feb 04 2020

Even-indexed rows have length n - 1; odd-indexed rows have length n - 2.
Conjecture: T(n, 1) = A083329(n - 1).
T(2n, 2n-1) = 2n, since the only derangement of 2n letters with 2n-1 descents is [2n, 2n-1,...,3,2,1].
The analogous sequence for permutations is T'(n, k) = k + 1.

			Triangle begins:
    2;
    5;
   11,    3,    4;
   23,   10,   17;
   47,  367,  173,  121,   6;
   95, 1431, 3973, 2513, 331;
  191,  567, 1395, 8011, 391, 761, 8.
T(4,1) = 11 because the derangements of four letters with one descent are
[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 11 as its numerator.

Cf. A000166, A083329, A219836.

Denominators are given by A332010.

descents(p)={sum(i=2, #p, p[i]Andrew Howroyd, Feb 02 2022

Terms a(27) and beyond from Andrew Howroyd, Feb 02 2022

A332010 Irregular triangle of denominators of the average value of the first letter over all derangements of {1, 2, ..., n} with k descents.

Original entry on oeis.org

1, 2, 4, 1, 1, 8, 3, 4, 16, 104, 40, 24, 1, 32, 392, 896, 480, 54, 64, 152, 308, 1496, 63, 108, 1, 128, 2276, 30384, 14410, 4315, 3024, 14, 256, 7340, 153400, 235252, 24766, 180416, 4984, 448, 1, 512, 23172, 365520, 1713160, 5794944, 3739512, 881152, 63840, 453

Offset: 2

Peter Kagey, Feb 04 2020

Even-indexed rows have length n - 1; odd-indexed rows have length n - 2.
Conjecture: T(n, 1) = 2^(n-2).
T(2n, 2n-1) = 1, since there is only one derangement of 2n letters with 2n-1 descents.
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).

			Triangle begins:
    1;
    2;
    4,    1,     1;
    8,    3,     4;
   16,  104,    40,    24,    1;
   32,  392,   896,   480,   54;
   64,  152,   308,  1496,   63,  108,  1;
  128, 2276, 30384, 14410, 4315, 3024, 14.
T(4,1) = 4 because the derangements of four letters with one descent are
[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.

Cf. A000166, A083329, A219836.

Numerators are given by A332009.

cp's OEIS Frontend

A332009 Irregular triangle of numerators of the average value of the first letter over all derangements of {1, 2, ..., n} with k descents.

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