A165434 -id:A165434 - cp's OEIS frontend

A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.

1, 2, 1, 5, 3, 1, 15, 16, 4, 1, 52, 139, 39, 5, 1, 203, 1750, 862, 81, 6, 1, 877, 29388, 35775, 4079, 150, 7, 1, 4140, 624889, 2406208, 507549, 15791, 256, 8, 1, 21147, 16255738, 238773109, 127126912, 5442547, 52450, 410, 9, 1, 115975, 504717929, 32867762616

Offset: 1

R. H. Hardin, Mar 30 2011

			Array begins:
========================================================================
n\k| 1  2   3      4         5            6            7               8
---+--------------------------------------------------------------------
1  | 1  2   5     15        52          203           877           4140
2  | 1  3  16    139      1750        29388        624889       16255738
3  | 1  4  39    862     35775      2406208     238773109    32867762616
4  | 1  5  81   4079    507549    127126912   55643064708 38715666455777
5  | 1  6 150  15791   5442547   4762077620 8738543204786
6  | 1  7 256  52450  46757209 135029200594
7  | 1  8 410 154279 335279744
8  | 1  9 625 411180
9  | 1 10 915
     ...
All solutions for 6 X 2
..1..1....1..1....1..0....1..1
..1..1....1..1....1..0....1..0
..1..0....1..1....1..0....1..0
..0..1....0..0....0..1....0..1
..0..0....0..0....0..1....0..1
..0..0....0..0....0..1....0..0

Andrew Howroyd, Table of n, a(n) for n = 1..181 (terms 1..69 from R. H. Hardin)

Rows 1..8 are A000110, A020554, A165434, A165435, A165436, A165437, A188393, A188394.
Columns 3..7 are A011863(n+1), A175707, A188389, A188390, A188391.
Main diagonal gives A188388.
Cf. A188445, A219727, A330942.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 12 2018

A322487 Number of (3*n) X n matrices with nonnegative integer entries and each column sum being 3 up to permutation of rows.

Original entry on oeis.org

1, 3, 31, 686, 27036, 1688360, 154703688, 19692332568, 3342458334775, 732812082630803, 202322386045180686, 68898094282978653925, 28443422251718020038049, 14029033632468285836567998, 8164217197799501761637725983, 5545466507405459243366712102466

Offset: 0

Andrew Howroyd, Dec 11 2018

nonn

Also number of multiset partitions of [1,1,1,2,2,2,...,n,n,n] into nonempty multisets. - Marko Riedel, Nov 29 2022

			a(1) = 3 because up to permutations of rows there are 3 column vectors with sum 3: [1, 1, 1], [2, 1, 0] and [3, 0, 0].

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=3 of A219727.

Cf. A165434, A358721.

A360037 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 2, 2, 2, ..., n, n, n] into k nonempty subsets, for 3 <= k <= 3n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 10, 13, 7, 3, 1, 1, 14, 92, 221, 249, 172, 81, 25, 6, 1, 1, 50, 872, 4277, 8806, 9840, 6945, 3377, 1206, 325, 65, 10, 1, 1, 186, 8496, 85941, 320320, 585960, 627838, 442321, 221475, 82985, 24038, 5496, 995, 140, 15, 1

Offset: 1

Marko Riedel, Jan 22 2023

A generalization of ordinary Stirling set numbers to multisets that contain some m instances each of n elements, here we have m=3.

			The triangular array starts:
[1]: 1;
[2]: 1,  1,  1,   1;
[3]: 1,  4, 10,  13,   7,   3,  1;
[4]: 1, 14, 92, 221, 249, 172, 81, 25, 6, 1;

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel, Rows 1 to 10 of triangle, flattened.
Marko Riedel, Documentation of the algorithm used in the Maple code.
Marko Riedel, Maple code for sequence by plain enumeration and the Polya Enumeration Theorem, by substitution and by recurrence.

Row sums are A165434.

Cf. A098233, A360038, A360039.

read "a360037maple":  # see link
A360037Row := n -> seq(T2(n, k, 3), k = 3..n*3): seq(A360037Row(n), n = 1..6);

A165435 Number of quad-coverings of a set.

Original entry on oeis.org

1, 1, 5, 81, 4079, 507549, 127126912, 55643064708, 38715666455777, 40095856807088486, 58901884724160709571, 118283825763578012358080, 315297447553856723754810322, 1089117884184737631305395668913, 4778248300569771278502378511288513

Offset: 0

Doron Zeilberger, Sep 18 2009

nonn

Analogous to A165434. See comments and references there.

Row 4 of A188392.

Cf. A165434.

A165436 Number of penta-coverings of a set.

Original entry on oeis.org

1, 1, 6, 150, 15791, 5442547, 4762077620, 8738543204786, 29476338150380267, 166824392081434701613, 1483193888081012069795493, 19709785177023094680286421698, 376477754511485610186131980496220, 10014420201202866850432679446597128835

Offset: 0

Doron Zeilberger, Sep 18 2009

nonn

Analogous to A165434. See references and comments there.

Row 5 of A188392.

Cf. A165434.

A175707 Number of ways to put n copies of 1,2,3,4 into sets.

Original entry on oeis.org

1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756, 351989816, 575711716, 921889652, 1447822620, 2233501928, 3389114724, 5064582169, 7461570579, 10848490675, 15579077786, 22115241763, 31054971635, 43166197978, 59427633555, 81077755892, 109673237289, 147158299390, 195946638641

Offset: 0

Thotsaporn Thanatipanonda, Dec 04 2010

nonn

Related to generalized Bell Numbers.
The n copies of each digit must be in different sets, and the sets must be nonempty.
Other definition: Number of ways to distribute n copies of 1,2,3,4 into an arbitrary number of (nonempty) sets. Due to the nature of sets, the same digit may not be several times in the same set.

			For n=1, the solution is the fourth term of Bell numbers A000110.
For n=2, one way to partition 2 copies of 1, 2 copies of 2, 2 copies of 3 and 2 copies of 4 is {1}{2}{34}{12}{34}. On the other hand {112}{34}{23}{4} is not allowed since the same numbers are in the same set {112}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls? (2009).
Doron Zeilberger, In How many ways can you reassemble several russian dolls?, arXiv:0909.3453 [math.CO], 2009.
Doron Zeilberger, BABUSHKAS; Local copy
Index entries for linear recurrences with constant coefficients, signature (7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1).

Cf. A000110, A011863, A020554, A165434.

a:= n-> (5382*n^11 +236808*n^10 +4643760*n^9 +53507520*n^8 +402098796*n^7 +2067612624*n^6 +7421736960*n^5 +18616942080*n^4 +32101468047*n^3 +36555545268*n^2 +25131098880*n +8024016000 +7016625*(-1)^n*n^3 +84199500*(-1)^n*n^2 +359251200*(-1)^n*n +538876800*(-1)^n) /(2^11*3^7*5^2*7*11) +5/3^6*(-1)^n * (sin(n*Pi/3)/sqrt(3)+ cos(n*Pi/3));
seq(a(n), n=0..40);
seq(SeqBrnDJ(n,4)[5], n=1..6); # using the Maple package BABUSHKAS (see links)

LinearRecurrence[{7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1}, {1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756}, 36] (* Jean-François Alcover, Nov 13 2018 *)

a(n) = (5382*n^11 +236808*n^10 +4643760*n^9 +53507520*n^8 +402098796*n^7 +2067612624*n^6 +7421736960*n^5 +18616942080*n^4 +32101468047*n^3 +36555545268*n^2 +25131098880*n +8024016000 +7016625*(-1)^n*n^3 +84199500*(-1)^n*n^2 +359251200*(-1)^n*n +538876800*(-1)^n) / (2^11*3^7*5^2*7*11) +5/3^6*(-1)^n * (sin(n*Pi/3)/sqrt(3) +cos(n*Pi/3)).
Recurrence: a(n) -7*a(n-1) +17*a(n-2) -8*a(n-3) -36*a(n-4) +60*a(n-5) -4*a(n-6) -56*a(n-7) +22*a(n-8) +22*a(n-9) +22*a(n-10) -56*a(n-11) -4*a(n-12) +60*a(n-13) -36*a(n-14) -8*a(n-15) +17*a(n-16) -7*a(n-17) +a(n-18) = 0.
G.f.: (x^10 +8*x^9 +51*x^8 +136*x^7 +252*x^6 +300*x^5 +252*x^4 +136*x^3 +51*x^2 +8*x+1) / ((x^2+x+1)*(x+1)^4*(x-1)^12).

A331389 Number of binary matrices with n distinct columns and any number of nonzero rows with 3 ones in every column and rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 3, 29, 666, 28344, 1935054, 193926796, 26892165502, 4946464286746, 1168900475263013, 346080409272270888, 125798338606148948325, 55204084562033205121607, 28834556615453989801860765, 17710828268156331289770544579, 12658784968736373972502731143309

Offset: 0

Andrew Howroyd, Jan 15 2020

nonn

The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows.

a(n) is the number of T_0 3-regular set multipartitions (multisets of sets) on an n-set.

			The a(2) = 3 matrices are:
   [1 0]   [1 1]   [1 1]
   [1 0]   [1 0]   [1 1]
   [1 0]   [1 0]   [1 0]
   [0 1]   [0 1]   [0 1]
   [0 1]   [0 1]
   [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=3 of A331126.

Cf. A165434.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A165434(k). - Andrew Howroyd, Jan 31 2020

A165437 The number of six-fold coverings of a set.

Original entry on oeis.org

1, 1, 7, 256, 52450, 46757209, 135029200594, 992649275638900, 15637211080656145509, 468371757818481922636261, 24491541042280069722306974244, 2097907014780540445950132842480077, 280173584370772226399860619059838944713, 56066456120812871209756615894736908159099853

Offset: 0

Doron Zeilberger, Sep 18 2009

nonn

Analogous to A165434. See comments and references there.

Row n=6 of A188392.

Cf. A165434.

a(11)-a(13) from Andrew Howroyd, Dec 10 2018

cp's OEIS Frontend

A188392 T(n,k) = number of (n*k) X k binary arrays with rows in nonincreasing order and n ones in every column.

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A322487 Number of (3*n) X n matrices with nonnegative integer entries and each column sum being 3 up to permutation of rows.

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A360037 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 2, 2, 2, ..., n, n, n] into k nonempty subsets, for 3 <= k <= 3n.

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A165435 Number of quad-coverings of a set.

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A165436 Number of penta-coverings of a set.

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A175707 Number of ways to put n copies of 1,2,3,4 into sets.

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A331389 Number of binary matrices with n distinct columns and any number of nonzero rows with 3 ones in every column and rows in nonincreasing lexicographic order.

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A165437 The number of six-fold coverings of a set.

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