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A319013 a(n) is the sum over each permutation of S_n of the least element of the descent set. 0
0, 1, 7, 37, 201, 1231, 8653, 69273, 623521, 6235291, 68588301, 823059733, 10699776673, 149796873591, 2246953104061, 35951249665201, 611171244308673, 11001082397556403, 209020565553571981, 4180411311071439981, 87788637532500240001, 1931350025715005280463 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(1) = 0 since the descent set of the identity permutation is empty.

Lim_{n->infinity} a(n)/n! = e - 1.

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 2011; see Section 1.4, pp. 38.

LINKS

Table of n, a(n) for n=1..22.

FORMULA

a(n) = Sum_{k=1..n-1} k^2*binomial(n, k+1)*(n - k - 1)!

a(n+1) = (n+1)*a(n) + n^2, with a(1) = 0.

a(n) = A002627(n) - n.

EXAMPLE

For n = 3, the least element of the descent set for each permutation in S_3 is given by the table:

+-------------+-------------+----------------------+

| permutation | descent set | least element (or 0) |

+-------------+-------------+----------------------+

| 123         | {}          | 0                    |

| 132         | {2}         | 2                    |

| 213         | {1}         | 1                    |

| 231         | {2}         | 2                    |

| 312         | {1}         | 1                    |

| 321         | {1,2}       | 1                    |

+-------------+-------------+----------------------+

Thus a(3) = 0 + 2 + 1 + 2 + 1 + 1 = 7.

MATHEMATICA

Table[Sum[k^2*Binomial[n, k + 1]*(n - k - 1)!, {k, 1, n - 1}], {n, 1, 15}]

CROSSREFS

Cf. A002627.

Sequence in context: A002807 A124610 A002683 * A126475 A274674 A255672

Adjacent sequences:  A319010 A319011 A319012 * A319014 A319015 A319016

KEYWORD

nonn

AUTHOR

Peter Kagey, Sep 07 2018

STATUS

approved

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Last modified September 20 20:12 EDT 2019. Contains 327247 sequences. (Running on oeis4.)