A101030
Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set.
Original entry on oeis.org
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
Offset: 1
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;
-
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
A344112
Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.
Original entry on oeis.org
1, 3, 12, 7, 56, 485, 15, 240, 4015, 65280, 31, 992, 32525, 1047552, 33551307, 63, 4032, 261415, 16773120, 1073726199, 68719430080, 127, 16256, 2094965, 268419072, 34359660243, 4398046231168, 562949952597769, 255, 65280, 16770655, 4294901760, 1099511237151
Offset: 1
T(2,2) = (number of relations) - (number of functions) = 2^4 - 4 = 12.
Triangle T(n,k) begins:
1;
3, 12;
7, 56, 485;
15, 240, 4015, 65280;
31, 992, 32525, 1047552, 33551307;
-
Column[Table[2^(n*k) - k^n, {n, 10}, {k, n}], Center]
A344114
a(n) = 2^(n^2) - n!.
Original entry on oeis.org
1, 14, 506, 65512, 33554312, 68719476016, 562949953416272, 18446744073709511296, 2417851639229258349049472, 1267650600228229401496699576576, 2658455991569831745807614120520772352, 22300745198530623141535718272648361026978816, 748288838313422294120286634350736906063831234982912
Offset: 1
a(1) = 2^(1^2) - 1! = 1;
a(2) = 2^(2^2) - 2! = 14;
a(3) = 2^(3^2) - 3! = 506.
A344115
Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.
Original entry on oeis.org
1, 2, 14, 5, 58, 506, 12, 244, 4072, 65512, 27, 1004, 32708, 1048456, 33554312, 58, 4066, 262024, 16776856, 1073741104, 68719476016, 121, 16342, 2096942, 268434616, 34359735848, 4398046506064, 562949953416272, 248, 65480, 16776880, 4294965616, 1099511621056
Offset: 1
For T(2,2): the number of relations is 2^4 and the number of one-to-one functions is 2, so 2^4 - 2 = 14 and thus T(2,2) = 14.
Triangle T(n,k) begins:
1;
2, 14;
5, 58, 506;
12, 244, 4072, 65512;
27, 1004, 32708, 1048456, 33554312;
-
Table[2^(n*k) - k!/(k - n)!, {k, 10}, {n, k}] // Flatten
A344116
Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not onto functions.
Original entry on oeis.org
1, 3, 14, 7, 58, 506, 15, 242, 4060, 65512, 31, 994, 32618, 1048336, 33554312, 63, 4034, 261604, 16775656, 1073740024, 68719476016, 127, 16258, 2095346, 268427056, 34359721568, 4398046495984, 562949953416272, 255, 65282, 16771420, 4294926472, 1099511501776, 281474976519136, 72057594037786816, 18446744073709511296
Offset: 1
For T(2,2), the number of relations is 2^4 and the number of onto functions is 2, so 2^4 - 2 = 14.
Triangle T(n,k) begins:
1
3 14
7 58 506
15 242 4060 65512
31 994 32618 1048336 33554312
-
TableForm[Table[2^(n*k) - Sum[Binomial[k, k - i] (k - i)^n*(-1)^i, {i, 0, k}], {n, 5}, {k, n}]]
-
T(n,k) = 2^(n*k) - k!*stirling(n, k, 2); \\ Michel Marcus, Jun 26 2021
A347034
Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.
Original entry on oeis.org
0, 0, 2, 0, 3, 21, 0, 4, 40, 232, 0, 5, 65, 505, 3005, 0, 6, 96, 936, 7056, 45936, 0, 7, 133, 1561, 14287, 112609, 818503, 0, 8, 176, 2416, 26048, 241984, 2056832, 16736896, 0, 9, 225, 3537, 43929, 470961, 4601529, 42683841, 387057609, 0, 10, 280, 4960, 69760, 848800
Offset: 1
For T(2,3): the number of functions is 3^2 and the number of one-to-one functions is 6, so 3^2 - 6 = 3 and thus T(2,3) = 3.
Triangle T(n,k) begins:
k=1 k=2 k=3 k=4 k=5 k=6
n=1: 0 0 0 0 0 0
n=2: 2 3 4 5 6
n=3: 21 40 65 96
n=4: 232 505 936
n=5: 3005 7056
n=6: 45936
Cf.
A000312,
A002416,
A036679,
A068424,
A089072,
A101030,
A199656,
A344110,
A344112,
A344113,
A344114,
A344115,
A344116.
-
A347034 := proc(n,k)
k^n-k!/(k-n)! ;
end proc:
seq(seq(A347034(n,k),n=1..k),k=1..12) ; # R. J. Mathar, Jan 12 2023
-
Table[k^n - k!/(k - n)!, {k, 12}, {n, k}] // Flatten
-
T(n,k) = k^n - k!/(k - n)!;
row(k) = vector(k, i, T(i, k)); \\ Michel Marcus, Oct 01 2021
Showing 1-6 of 6 results.
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