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A101030 Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set. 6
0, 0, 2, 0, 2, 21, 0, 2, 45, 232, 0, 2, 93, 784, 3005, 0, 2, 189, 2536, 13825, 45936, 0, 2, 381, 7984, 61325, 264816, 818503, 0, 2, 765, 24712, 264625, 1488096, 5623681, 16736896, 0, 2, 1533, 75664, 1119005, 8172576, 38025127, 132766208, 387057609, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..46.

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.2(v).

D. P. Walsh, A note on non-surjective functions from [n] to [k].

FORMULA

T(n,k) = A089072(n,k) - A019538(n,k).

T(n,k) = Sum_{j=1..k} (-1)^(j-1)*C(k,j)*(k-j)^n. - Dennis P. Walsh, Apr 13 2016

T(n,k) = k^n - k!*Stirling2(n,k). - Dennis P. Walsh, Apr 13 2016

EXAMPLE

T(3,3) = #(functions into) - #(functions onto) = 3^3 - 6 = 21

Triangle T(n,k) begins:

0,

0, 2;

0, 2, 21;

0, 2, 45, 232;

0, 2, 93, 784, 3005;

0, 2, 189, 2536, 13825, 45936;

0, 2, 381, 7984, 61325, 264816, 818503;

0, 2, 765, 24712, 264625, 1488096, 5623681, 16736896;

0, 2, 1533, 75664, 1119005, 8172576, 38025127, 132766208, 387057609;

MAPLE

T:=(n, k)->sum((-1)^(j-1)*binomial(k, j)*(k-j)^n, j=1..k);

seq(seq(T(n, k), k=1..n), n=1..15); # Dennis P. Walsh, Apr 13 2016

CROSSREFS

Cf. A199656, A036679 (diagonal).

Sequence in context: A285152 A077184 A077183 * A093857 A056949 A346235

Adjacent sequences: A101027 A101028 A101029 * A101031 A101032 A101033

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Nov 26 2004

EXTENSIONS

Offset corrected from 0 to 1 by Dennis P. Walsh, Apr 13 2016

STATUS

approved

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Last modified November 27 03:04 EST 2022. Contains 358362 sequences. (Running on oeis4.)