| Displaying 1-8 of 8 results found.
page
1
Triangle T(n,k) of number of minimal 3-covers of an unlabeled n+3-set that cover k points of that set uniquely (k=3,..,n+3).
+10
10
1, 2, 1, 4, 3, 2, 7, 7, 6, 3, 11, 13, 14, 9, 4, 16, 22, 26, 21, 13, 5, 23, 34, 44, 40, 31, 17, 7, 31, 50, 68, 68, 59, 41, 23, 8, 41, 70, 100, 106, 101, 79, 55, 28, 10, 53, 95, 140, 157, 158, 136, 106, 68, 35, 12, 67, 125, 190, 221, 234, 214, 182, 132, 85, 42, 14, 83, 161
FORMULA
T(n, k) = b(n, k)-b(n-1, k); b(n, k) = coefficient of x^k in x^3/6*(Z(S_n; 5+3*x, 5+3*x^2, ...)+3*Z(S_n; 3+x, 5+3*x^2, 3+x^3, 5+3*x^4, ...)+2*Z(S_n; 2, 2, 5+3*x^3, 2, 2, 5+3*x^6, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
EXAMPLE
[1], [2, 1], [4, 3, 2], [7, 7, 6, 3], ...
There are 7 minimal 3-covers of an unlabeled 6-set that cover 3 points of that set uniquely: {{1}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 4, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}.
Triangle T(n,k) of numbers of minimal 5-covers of an unlabeled n+5-set that cover k points of that set uniquely (k=5,..,n+5).
+10
9
1, 4, 1, 19, 7, 2, 91, 46, 16, 3, 436, 279, 115, 28, 5, 1991, 1563, 740, 221, 49, 7, 8651, 7978, 4309, 1524, 405, 75, 10, 35354, 37290, 22604, 9272, 2875, 659, 115, 13, 135617, 159948, 107584, 50058, 17840, 4866, 1042, 163, 18, 488312, 633211
FORMULA
T(n, k)=b(n, k)-b(n-1, k); b(n, k)=coefficient of x^k in (x^5/5!)*(Z(S_n; 27+5*x, 27+5*x^2, ...)+10*Z(S_n; 13+3*x, 27+5*x^2, 13+3*x^3, 27+5*x^4, ...)+15*Z(S_n; 7+x, 27+5*x^2, 7+x^3, 27+5*x^4, ...)+20*Z(S_n; 6+2*x, 6+2*x^2, 27+5*x^3, 6+2*x^4, 6+2*x^5, 27+5*x^6, ...)+20*Z(S_n; 4, 6+2*x^2, 13+3*x^3, 6+2*x^4, 4, 27+5*x^6, 4, 6+2*x^8, 13+3*x^9, 6+2*x^10, 4, 27+5*x^12, ...)+30*Z(S_n; 3+x, 7+x^2, 3+x^3, 27+5*x^4, 3+x^5, 7+x^6, 3+x^7, 27+5*x^8, ...)+24*Z(S_n; 2, 2, 2, 2, 27+5*x^5, 2, 2, 2, 2, 27+5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
EXAMPLE
[1], [4, 1], [19, 7, 2], [91, 46, 16, 3], [436, 279, 115, 28, 5], ...; there are 46 minimal 5-covers of an unlabeled 8-set that cover 6 points of that set uniquely.
CROSSREFS
Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965, A057966(labeled case), A057967.
Number of 4 X n binary matrices without unit columns up to row and column permutations.
+10
8
1, 4, 14, 44, 127, 335, 830, 1931, 4258, 8943, 17984, 34765, 64873, 117220, 205718, 351552, 586348, 956393, 1528350, 2396631, 3693123, 5599550, 8363304, 12317274, 17904795, 25710327, 36497466, 51255153, 71253960, 98113791, 133885404, 181147299, 243121170, 323807952, 428148174
COMMENTS
A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 4-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).
FORMULA
1/24*(Z(S_n; 12, 12, ...) + 8*Z(S_n; 3, 3, 12, 3, 3, 12, ...) + 6*Z(S_n; 6, 12, 6, 12, ...) + 3*Z(S_n; 4, 12, 4, 12, ...) + 6*Z(S_n; 2, 4, 2, 12, 2, 4, 2, 12, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f. : 1/24*(1/(1 - x)^12 + 8/(1 - x)^3/(1 - x^3)^3 + 6/(1 - x)^6/(1 - x^2)^3 + 3/(1 - x)^4/(1 - x^2)^4 + 6/(1 - x)^2/(1 - x^2)/(1 - x^4)^2).
PROG
(PARI) x='x+O('x^66); Vec(1/24*(1/(1-x)^12 + 8/(1-x)^3/(1-x^3)^3 + 6/(1-x)^6/(1-x^2)^3 + 3/(1-x)^4/(1-x^2)^4 + 6/(1-x)^2/(1-x^2)/(1-x^4)^2)) \\ Joerg Arndt, May 21 2013
Triangle T(n,k) of numbers of minimal 4-covers of an unlabeled n+4-set that cover k points of that set uniquely (k=4,..,n+4).
+10
4
1, 3, 1, 10, 5, 2, 30, 21, 11, 3, 83, 75, 49, 18, 5, 208, 231, 177, 84, 30, 6, 495, 636, 554, 318, 143, 42, 9, 1101, 1603, 1540, 1023, 543, 210, 62, 11, 2327, 3737, 3907, 2904, 1759, 822, 311, 82, 15, 4685, 8163, 9153, 7470, 5012, 2706, 1219, 423, 111, 18, 9041
FORMULA
T(n, k) = b(n, k)-b(n-1, k); b(n, k) = coefficient of x^k in x^4/24*(Z(S_n; 12 + 4*x, 12 + 4*x^2, ...) + 8*Z(S_n; 3 + x, 3 + x^2, 12 + 4*x^3, 3 + x^4, 3 + x^5, 12 + 4*x^6, ...) + 6*Z(S_n; 6 + 2*x, 12 + 4*x^2, 6 + 2*x^3, 12 + 4*x^4, ...)
+ 3*Z(S_n; 4, 12 + 4*x^2, 4, 12 + 4*x^4, ...) + 6*Z(S_n; 2, 4, 2, 12 + 4*x^4, 2, 4, 2, 12 + 4*x^8, ...)), where Z(S_n; x_1, x_2, ..., x_n) is the cycle index of the symmetric group S_n of degree n.
EXAMPLE
[1], [3, 1], [10, 5, 2], [30, 21, 11, 3], [83, 75, 49, 18], ...; there are 5 minimal 4-covers of an unlabeled 6-set that cover 5 points of that set uniquely.
CROSSREFS
Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965(labeled case), A057966, A057968.
Number of 5 X n binary matrices with 3 unit columns up to row and column permutations.
+10
4
3, 31, 252, 1776, 11048, 61106, 303664, 1368844, 5651241, 21559133, 76613440, 255411923, 803771681, 2400633464, 6837010458, 18644075466, 48855805143, 123415815229, 301386128354, 713271875603, 1639572164669, 3667859207856
COMMENTS
A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 8 points of that set uniquely (if offset is 8).
FORMULA
Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) + 20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f. : x^3/120*(35/(1 - x^1)^27 + 130/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 100/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).
5 x n binary matrices without unit columns up to row and column permutations.
+10
3
1, 5, 24, 115, 551, 2542, 11193, 46547, 182164, 670476, 2325506, 7624434, 23716419, 70253721, 198905506, 540079754, 1410786483, 3555443969, 8667153126, 20484365167, 47037898503, 105143200252, 229178029000
COMMENTS
A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5-covers of an unlabeled n-set that cover 5 points of that set uniquely (if offset is 5).
FORMULA
a(n)=(1/5!)*(Z(S_n; 27, 27, ...) + 10*Z(S_n; 13, 27, 13, 27, ...) + 15*Z(S_n; 7, 27, 7, 27, ...) + 20*Z(S_n; 6, 6, 27, 6, 6, 27, ...) + 20*Z(S_n; 4, 6, 13, 6, 4, 27, 4, 6, 13, 6, 4, 27, ...) + 30*Z(S_n; 3, 7, 3, 27, 3, 7, 3, 27, ...) + 24*Z(S_n; 2, 2, 2, 2, 27, 2, 2, 2, 2, 27, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f. : 1/120*(1/(1 - x^1)^27 + 10/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 20/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5 + 24/(1 - x^1)^2/(1 - x^5)^5).
5 x n binary matrices with 1 unit column up to row and column permutations.
+10
3
1, 8, 54, 333, 1896, 9874, 47164, 207112, 840323, 3168506, 11170331, 37034409, 116095018, 345785753, 982835676, 2676217504, 7005306389, 17681946594, 43153532167, 102080966243, 234565062960, 524594120393, 1143910860870
COMMENTS
A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 6 points of that set uniquely (if offset is 6).
FORMULA
Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) + 20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f.: x/120*(5/(1 - x^1)^27 + 30/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 40/(1 - x^1)^6/(1 - x^3)^7 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).
CROSSREFS
Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057970 - A057972, A057969, A057971, A057972.
Number of 5 x n binary matrices with 2 unit columns up to row and column permutations.
+10
2
2, 18, 133, 873, 5182, 27786, 135370, 602454, 2466628, 9358497, 33134431, 110184932, 346141949, 1032550097, 2938104492, 8006865684, 20971632456, 52958252851, 129291697111, 305924724070, 703108665327, 1572722761341
COMMENTS
A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 7 points of that set uniquely (if offset is 7).
FORMULA
Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) +
20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)),
where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f.: x^2/120*(15/(1 - x^1)^27 + 70/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 60/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).
CROSSREFS
Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963- A057968, A057970- A057972, A057969, A057970, A057972.
Search completed in 0.007 seconds
| 