cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A057968 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).

Original entry on oeis.org

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
Offset: 0

Views

Author

Vladeta Jovovic, Oct 17 2000

Keywords

Comments

Row sums give A005785.

Examples

			[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.

Links

Crossrefs

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965, A057966(labeled case), A057967.

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.

A057965 Triangle T(n,k) of number of minimal 4-covers of a labeled n-set that cover k points of that set uniquely (k=4,..,n).

Original entry on oeis.org

1, 55, 10, 1815, 660, 65, 46585, 25410, 5005, 350, 1024870, 745360, 220220, 30800, 1701, 20292426, 18447660, 7267260, 1524600, 168399, 7770, 372027810, 405848520, 199849650, 55902000, 9261945, 854700, 34105, 6430766430
Offset: 4

Views

Author

Vladeta Jovovic, Oct 17 2000

Keywords

Comments

Row sums give A016111.

Examples

			[1], [55, 10], [1815, 660, 65], [46585, 25410, 5005, 350], ...; there are 1815 minimal 4-covers of a labeled 6-set that cover 4 points of that set uniquely.

Links

Crossrefs

Cf. A035347, A057669, A057963, A057964, A057966, A057967(unlabeled case), A057968.

Formula

Number of minimal m-covers of a labeled n-set that cover k points of that set uniquely is C(n, k)*S(k, m)*(2^m-m-1)^(n-k), where S(k, m) are Stirling numbers of the second kind.

A057964 Triangle T(n,k) of number of minimal 3-covers of a labeled n-set that cover k points of that set uniquely (k=3,..,n).

Original entry on oeis.org

1, 16, 6, 160, 120, 25, 1280, 1440, 600, 90, 8960, 13440, 8400, 2520, 301, 57344, 107520, 89600, 40320, 9632, 966, 344064, 774144, 806400, 483840, 173376, 34776, 3025, 1966080, 5160960, 6451200, 4838400, 2311680, 695520, 121000, 9330
Offset: 3

Views

Author

Vladeta Jovovic, Oct 17 2000

Keywords

Comments

Row sums give A003468.

Examples

			[1], [16, 6], [160, 120, 25], [1280, 1440, 600, 90], ...; There are 305=160+120+25 minimal 3-covers of a labeled 5-set.

Links

Crossrefs

Cf. A035347, A057669 (unlabeled case), A057963, A057965-A057968.

Formula

Number of minimal m-covers of a labeled n-set that cover k points of that set uniquely is C(n, k)*S(k, m)*(2^m-m-1)^(n-k), where S(k, m) are Stirling numbers of the second kind.

A057966 Triangle T(n,k) of number of minimal 5-covers of a labeled n-set that cover k points of that set uniquely (k=5,..,n).

Original entry on oeis.org

1, 156, 15, 14196, 2730, 140, 984256, 283920, 29120, 1050, 57578976, 22145760, 3407040, 245700, 6951, 2994106752, 1439474400, 295276800, 31941000, 1807260, 42525, 142719088512, 82337935680, 21112291200, 3045042000, 258438180
Offset: 5

Views

Author

Vladeta Jovovic, Oct 17 2000

Keywords

Comments

Row sums give A046166.

Examples

			[1], [156, 15], [14196, 2730, 140], [984256, 283920, 29120, 1050], ...; there are 15 minimal 5-covers of a labeled 6-set that cover 6 points of that set uniquely.

Links

Crossrefs

Cf. A035347, A057669, A057963-A057965, A057967, A057968(unlabeled case).

Formula

Number of minimal m-covers of a labeled n-set that cover k points of that set uniquely is C(n, k)*S(k, m)*(2^m-m-1)^(n-k), where S(k, m) are Stirling numbers of the second kind.

A057967 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).

Original entry on oeis.org

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
Offset: 0

Views

Author

Vladeta Jovovic, Oct 17 2000

Keywords

Comments

Row sums give A005784.

Examples

			[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.

Links

Crossrefs

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965(labeled case), A057966, A057968.

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.

A057972 Number of 5 X n binary matrices with 3 unit columns up to row and column permutations.

Original entry on oeis.org

3, 31, 252, 1776, 11048, 61106, 303664, 1368844, 5651241, 21559133, 76613440, 255411923, 803771681, 2400633464, 6837010458, 18644075466, 48855805143, 123415815229, 301386128354, 713271875603, 1639572164669, 3667859207856
Offset: 3

Views

Author

Vladeta Jovovic, Oct 21 2000

Keywords

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).

Links

Crossrefs

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057969 - A057971.

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).

A057969 5 x n binary matrices without unit columns up to row and column permutations.

Original entry on oeis.org

1, 5, 24, 115, 551, 2542, 11193, 46547, 182164, 670476, 2325506, 7624434, 23716419, 70253721, 198905506, 540079754, 1410786483, 3555443969, 8667153126, 20484365167, 47037898503, 105143200252, 229178029000
Offset: 0

Views

Author

Vladeta Jovovic, Oct 20 2000

Keywords

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).

Links

Crossrefs

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963-A057968, A057970-A057972.

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).

A057970 5 x n binary matrices with 1 unit column up to row and column permutations.

Original entry on oeis.org

1, 8, 54, 333, 1896, 9874, 47164, 207112, 840323, 3168506, 11170331, 37034409, 116095018, 345785753, 982835676, 2676217504, 7005306389, 17681946594, 43153532167, 102080966243, 234565062960, 524594120393, 1143910860870
Offset: 1

Views

Author

Vladeta Jovovic, Oct 21 2000

Keywords

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).

Links

Crossrefs

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057970 - A057972, A057969, A057971, A057972.

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).

A057971 Number of 5 x n binary matrices with 2 unit columns up to row and column permutations.

Original entry on oeis.org

2, 18, 133, 873, 5182, 27786, 135370, 602454, 2466628, 9358497, 33134431, 110184932, 346141949, 1032550097, 2938104492, 8006865684, 20971632456, 52958252851, 129291697111, 305924724070, 703108665327, 1572722761341
Offset: 2

Views

Author

Vladeta Jovovic, Oct 21 2000

Keywords

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).

Links

Crossrefs

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963-A057968, A057970-A057972, A057969, A057970, A057972.

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).

A280752 Numerators of triangle related to enumeration of minimal 2-covers of a labeled n-set.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 10, 1, 1, 2, 865, 71, 1, 1, 5, 2630, 1427, 181, 1, 1, 3, 163133, 306553, 36667, 145, 1, 1, 7, 3368938, 129115655, 46958822, 43662, 4036, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jan 15 2017

Keywords

Examples

			Triangle begins:
1,
1/3,   1/3,
1/7,   1/2,      1/7,
1/15,  3/5,     10/21,         1/15,
1/31,  2/3,    865/651,       71/186,       1/31,
1/63,  5/7,   2630/651,     1427/651,     181/651,     1/63,
1/127, 3/4, 163133/11811, 306553/15748, 36667/11811, 145/762, 1/127,
...

Links

Crossrefs

Cf. A022166, A057963, A280753.
Showing 1-10 of 11 results. Next

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