A330942 -id:A330942 - cp's OEIS frontend

A331567 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column.

1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 1, 13, 6, 0, 1, 1, 75, 120, 0, 0, 1, 1, 541, 6174, 1104, 0, 0, 1, 1, 4683, 449520, 413088, 5040, 0, 0, 1, 1, 47293, 49686726, 329520720, 18481080, 0, 0, 0, 1, 1, 545835, 7455901320, 491236986720, 179438982360, 522481680, 0, 0, 0, 1

Offset: 0

Andrew Howroyd, Jan 20 2020

			Array begins:
===============================================================
n\k | 0 1 2    3          4              5                6
----+----------------------------------------------------------
  0 | 1 1 1    1          1              1                1 ...
  1 | 1 1 3   13         75            541             4683 ...
  2 | 1 0 6  120       6174         449520         49686726 ...
  3 | 1 0 0 1104     413088      329520720     491236986720 ...
  4 | 1 0 0 5040   18481080   179438982360 3785623968170400 ...
  5 | 1 0 0    0  522481680 70302503250720 ...
  6 | 1 0 0    0 7875584640 ...
  ...
The A(2,2) = 6 matrices are:
   [1 1]  [1 1]  [1 0]  [1 0]  [0 1]  [0 1]
   [1 0]  [0 1]  [1 1]  [0 1]  [1 1]  [1 0]
   [0 1]  [1 0]  [0 1]  [1 1]  [1 0]  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..209

Rows n=1..3 are A000670, A331640, A331641.
Column k=5 is A331642.
Cf. A188445, A330942, A331568, A331569, A331571, A331643.

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+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

A(n,k) = 0 for k > 0, n > 2^(k-1).
A(2^(k-1), k) = (2^k-1)! for k > 0.
A331643(n) = Sum_{d|n} A(n/d, d).

A331510 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of distinct nonzero rows with n ones in every column up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 3, 1, 0, 1, 1, 5, 4, 0, 0, 1, 1, 7, 12, 3, 0, 0, 1, 1, 11, 36, 23, 1, 0, 0, 1, 1, 15, 124, 191, 30, 0, 0, 0, 1, 1, 22, 412, 2203, 837, 23, 0, 0, 0, 1, 1, 30, 1500, 31313, 41664, 2688, 12, 0, 0, 0, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

			Array begins:
=================================
n\k | 0 1 2 3  4   5    6   7
----+----------------------------
  0 | 1 1 1 1  1   1    1   1 ...
  1 | 1 1 2 3  5   7   11  15 ...
  2 | 1 0 1 4 12  36  124 412 ...
  3 | 1 0 0 3 23 191 2203 ...
  4 | 1 0 0 1 30 837 ...
  5 | 1 0 0 0 23 ...
  ...
The A(2,3) = 4 matrices are:
  [1 1 1]  [1 1 0]  [1 1 1]  [1 1 0]
  [1 0 0]  [1 0 1]  [1 1 0]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 0 1]  [0 1 1]
  [0 0 1]  [0 0 1]

Rows n=1..3 are A000041, A331717, A331718.
Column k=5 is A331719.
Cf. A188445, A330942, A331461, A331508, A331509.

A(n,k) = 0 for k > 0, n > 2^(k-1).

A(n,k) = A(2^(k-1) - n, k) for k > 0, n <= 2^(k-1).

a(58)-a(65) from Andrew Howroyd, Feb 08 2020

A331278 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 12, 4, 1, 0, 1, 124, 124, 8, 1, 0, 1, 1800, 10596, 1280, 16, 1, 0, 1, 33648, 1764244, 930880, 13456, 32, 1, 0, 1, 769336, 484423460, 1849386640, 85835216, 143808, 64, 1, 0, 1, 20796960, 198461691404, 7798297361808, 2098356708016, 8206486848, 1556416, 128, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.

A(n,k) is the number of n-uniform k-block sets of multisets.

			Array begins:
====================================================================
n\k | 0  1      2          3                4                   5
----+---------------------------------------------------------------
  0 | 1  1      0          0                0                   0 ...
  1 | 1  1      1          1                1                   1 ...
  2 | 1  2     12        124             1800               33648 ...
  3 | 1  4    124      10596          1764244           484423460 ...
  4 | 1  8   1280     930880       1849386640       7798297361808 ...
  5 | 1 16  13456   85835216    2098356708016  140094551934813712 ...
  6 | 1 32 143808 8206486848 2516779512105152 ...
  ...
The A(2,2) matrices are:
  [1 0]  [1 0]  [1 0]  [2 0]  [1 1]  [1 0]
  [1 0]  [0 1]  [0 1]  [0 1]  [1 0]  [1 1]
  [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [0 1]
  [0 1]  [0 1]  [1 0]
.
  [1 0]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]
  [1 0]  [0 2]  [0 1]  [0 1]  [0 2]  [1 2]
  [0 2]  [1 0]  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=1..2 are A000012, A173219.
Columns k=0..2 are A000012, A011782, A331396.
The version with binary entries is A331277.
The version with not necessarily distinct columns is A331315.
Cf. A316674 (unrestricted version), A330942, A331395.

T(n,k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1,n),k)*sum(i=j, m, (-1)^(i-j)*binomial(i,j)))}

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331315(n, j).
A(n, k) = A011782(n) * A331277(n, k) for k > 0.
A331395(n) = Sum_{d|n} A(n/d, d).

A331509 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set-systems on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 3, 0, 0, 1, 0, 1, 6, 3, 0, 0, 1, 0, 1, 15, 19, 1, 0, 0, 1, 0, 1, 42, 141, 29, 0, 0, 0, 1, 0, 1, 109, 1571, 769, 23, 0, 0, 0, 1, 0, 1, 320

Offset: 0

Andrew Howroyd, Jan 18 2020

An n-regular set-system is a finite set of nonempty sets in which each element appears in n blocks.
A set-system is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of nonequivalent binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column up to permutation of rows and columns.

			Array begins:
=================================
n\k | 0 1 2 3  4   5    6   7
----+----------------------------
  0 | 1 1 0 0  0   0    0   0 ...
  1 | 1 1 1 1  1   1    1   1 ...
  2 | 1 0 1 3  6  15   42 109 ...
  3 | 1 0 0 3 19 141 1571 ...
  4 | 1 0 0 1 29 769 ...
  5 | 1 0 0 0 23 ...
  ...
The A(2,3) = 3 matrices are:
  [1 1 1]    [1 1 0]    [1 1 0]
  [1 0 0]    [1 0 1]    [1 0 1]
  [0 1 0]    [0 1 0]    [0 1 1]
  [0 0 1]    [0 0 1]

Row 2 appears to be A005368. Row 3 is A331716.

Cf. A330942, A331039, A331461, A331508, A331510.

A226994 Number of lattice paths from (0,0) to (n,n) consisting of steps U=(1,1), H=(1,0) and S=(0,1) such that the first step leaving the diagonal (if any) is an H step.

Original entry on oeis.org

1, 2, 7, 32, 161, 842, 4495, 24320, 132865, 731282, 4048727, 22523360, 125797985, 704966810, 3961924127, 22321190912, 126027618305, 712917362210, 4039658528935, 22924714957472, 130271906898721, 741188107113962, 4221707080583087, 24070622500965632

Offset: 0

Alois P. Heinz, Jun 26 2013

nonn

a(n) is also the n-th order truncated expansion in x and y of 1/(1-x*y+x+y) evaluated at x=1, y=1 (see Mathematica code). - Benedict W. J. Irwin, Oct 06 2016

			a(0) = 1: the empty path.
a(1) = 2: HS, U.
a(2) = 7: HHSS, HSHS, HSSH, HSU, HUS, UHS, UU.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=2 of A330942.

Cf. A001850 (unrestricted paths), A006318 (subdiagonal paths), A047665, A182626, A226995, A226996.

a:= proc(n) option remember; `if`(n<3, n*(2*n-1)+1,
     ((n-2)*(2*n-1) *a(n-3) -(7*n-4)*(2*n-3) *a(n-2)
      +(2*n-1)*(7*n-10) *a(n-1))/ (n*(2*n-3)))
    end:
seq(a(n), n=0..25);

Table[CoefficientList[Series[1/(1-x*y+x+y), {x, 0, n}, {y, 0, n}], z][[1]] /.x -> 1 /. y -> 1, {n, 0, 10}] (* Benedict W. J. Irwin, Oct 06 2016 *)

a(n) = 1/2 + pollegendre(n, 3)/2; \\ Michel Marcus, Oct 06 2016

G.f.: 1/(2-2*x) + 1/(2*sqrt(1-6*x+x^2)).
a(n) = A001850(n) - A047665(n).
a(n) = 1/2 + LegendreP(n, 3)/2. - Benedict W. J. Irwin, Oct 06 2016
a(n) ~ sqrt(3*sqrt(2) + 4) * (3 + 2*sqrt(2))^n / (4*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 07 2016
a(n) = Sum_{k=0..n} (-1)^k * A182626(k). - J. Conrad, Apr 08 2018
a(n) = 1 + Sum_{k=1..n} binomial(n,k)^2 * 2^(k-1). - Ilya Gutkovskiy, Nov 15 2021
a(n) = 1 + A047665(n). - Alois P. Heinz, Nov 15 2021

A331277 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 6, 1, 1, 0, 1, 62, 31, 1, 1, 0, 1, 900, 2649, 160, 1, 1, 0, 1, 16824, 441061, 116360, 841, 1, 1, 0, 1, 384668, 121105865, 231173330, 5364701, 4494, 1, 1, 0, 1, 10398480, 49615422851, 974787170226, 131147294251, 256452714, 24319, 1, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.

A(n,k) is the number of labeled n-uniform hypergraphs with k edges and no isolated vertices. When n=2 these objects are graphs.

			Array begins:
====================================================================
n\k | 0 1    2         3              4            5           6
----+---------------------------------------------------------------
  0 | 1 1    0         0              0            0           0 ...
  1 | 1 1    1         1              1            1           1 ...
  2 | 1 1    6        62            900        16824      384668 ...
  3 | 1 1   31      2649         441061    121105865 49615422851 ...
  4 | 1 1  160    116360      231173330 974787170226 ...
  5 | 1 1  841   5364701   131147294251 ...
  6 | 1 1 4494 256452714 78649359753286 ...
  ...
The A(2,2) = 6 matrices are:
   [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]
   [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 1]
   [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [1 1]
   [0 1]  [0 1]  [1 0]

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=1..3 are A000012, A121251, A136245.
Columns k=0..3 are A000012, A000012, A047665, A137219.
The version with nonnegative integer entries is A331278.
The version with not necessarily distinct columns is A330942.
Cf. A262809 (unrestricted version), A331315, A331639.

T(n,k)={my(m=n*k); sum(j=0, m, binomial(binomial(j,n), k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}

A(n, k) = Sum_{j=0..n*k} binomial(binomial(j,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A262809(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A330942(n, j).
A331639(n) = Sum_{d|n} A(n/d, d).

A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 5, 16, 17, 140, 65, 1395, 2969, 22176, 1025, 1050766, 4097, 13010328, 128268897, 637598438, 65537, 64864962683, 262145, 1676258452736, 28683380484257, 24908619669860, 4194305, 30567710172480050, 8756434134071649, 62128557507554504, 21271147396968151093

Offset: 1

Andrew Howroyd, Jan 23 2020

nonn

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
From Gus Wiseman, Apr 03 2025: (Start)
Also the number of multiset partitions such that (1) the blocks together cover an initial interval of positive integers, (2) the blocks are sets of a common size, and (3) the block-sizes sum to n. For example, the a(1) = 1 through a(4) = 16 multiset partitions are:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{1,2}}
         {{1},{2}}  {{1},{1},{2}}  {{1,2},{1,3}}
                    {{1},{2},{2}}  {{1,2},{2,3}}
                    {{1},{2},{3}}  {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{1,3},{2,4}}
                                   {{1,4},{2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{2},{3}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A330942, A331639.
For constant instead of strict blocks we have A034729.
Without equal sizes we have A116540 (normal set multipartitions).
Without strict blocks we have A317583.
For distinct instead of equal sizes we have A382428, non-strict blocks A326517.
For equal sums instead of sizes we have A382429, non-strict blocks A326518.
Normal multiset partitions: A255903, A255906, A317532, A382203, A382204, A382216.
Cf. A000110, A000670, A007716, A034691, A035310, A050320, A050326, A116539, A306319, A381718.

a(n) = Sum_{d|n} A330942(n/d, d).

a(p) = 2^(p-1) + 1 for prime p.

A121316 Unlabeled version of A055203: number of different relations between n intervals (of nonzero length) on a line, up to permutation of intervals.

Original entry on oeis.org

1, 1, 7, 75, 1105, 20821, 478439, 12977815, 405909913, 14382249193, 569377926495, 24908595049347, 1193272108866953, 62128556769033261, 3493232664307133871, 210943871609662171055, 13615857409567572389361, 935523911378273899335537

Offset: 0

Goran Kilibarda and Vladeta Jovovic, Aug 25 2006

nonn

Also number of labeled multigraphs without isolated vertices and with n edges.

Nathaniel Johnston, Table of n, a(n) for n = 0..125
A. N. Bhavale, B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.

Row n=2 of A330942.

Cf. A055203, A121251, A104209.

seq(sum(binomial(k*(k-1)/2+n-1,n)/2^(k+1),k=0..infinity),n=0..20);
with(combinat): A121316:=proc(n) return (1/n!)*add(abs(stirling1(n,k))*A055203(k),k=0..n): end: seq(A121316(n),n=0..20); # Nathaniel Johnston, Apr 28 2011

Table[Sum[Binomial[k*(k-1)/2+n-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Mar 15 2014 *)

a(n) = {sum(j=0, 2*n, binomial(binomial(j,2)+n-1, n) * sum(i=j, 2*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A055203(k).
a(n) = Sum_{k>=0} binomial(k*(k-1)/2+n-1,n)/2^(k+1).
a(n) ~ n^n * 2^(n-1 + log(2)/4) / (exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Mar 15 2014
a(n) = Sum_{j=0..2*n} binomial(binomial(j,2)+n-1, n) * (Sum_{i=j..2*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.

Original entry on oeis.org

1, 4, 75, 2712, 116681, 5366384, 256461703, 12582521536, 629390010177, 31955248465164, 1641724961412515, 85159811886281576, 4452782349821587705, 234393562420377364008, 12409423916987553634575, 660253088667255226947072

Offset: 0

Vladeta Jovovic, Mar 06 2008, Mar 16 2008

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=3 of A330942.

Cf. A047665, A137219.

A126086 := proc(n) local x,y,z ; coeftayl(coeftayl(coeftayl(1/(1-x-y-z-x*y-x*z-y*z-x*y*z),z=0,n),y=0,n),x=0,n) ; end: A001850 := proc(n) local k ; add(binomial(n,k)*binomial(n+k,k),k=0..n) ; end: A137220 := proc(n) (A126086(n)+3*A001850(n)+2)/6 ; end: seq(A137220(n),n=0..30) ; # R. J. Mathar, Apr 01 2008

T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];
Table[T[n, 3], {n, 0, 15}] (* Jean-François Alcover, Apr 10 2020, after Andrew Howroyd *)

a(n) = {sum(j=0, 3*n, binomial(binomial(j,n)+2, 3) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

@CachedFunction
def A137220(n): return round( -sum( binomial(-binomial(j, n), 3)/2^(j+1) for j in (0..500) ) )
[A137220(n) for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = -Sum_{m>=0} binomial(-binomial(m,n),3)/2^(m+1).
a(n) = A137219(n) + A001850(n). - R. J. Mathar, Apr 01 2008
a(n) = Sum_{j=0..3*n} binomial(binomial(j,n)+2, 3) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

More terms from R. J. Mathar, Apr 01 2008

A136246 a(n) = (1/n!)Sum_{k=0..n} (-1)^(n-k)Stirling1(n,k)*A062208(k).

Original entry on oeis.org

1, 1, 32, 2712, 449102, 122886128, 50225389432, 28670796914144, 21789885975738524, 21271115441652577064, 25938193213744579451420, 38638907727108476424404864, 69044758685363149615280762608, 145768622491129079115419544343808, 358961215083489204505055286181798208

Offset: 0

Vladeta Jovovic, Mar 16 2008

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

Row n=3 of A330942.

Cf. A062208, A121316.

A000629 := proc(n) local k ; sum( k^n/2^k,k=0..infinity) ; end: A062208 := proc(n) option remember ; local a,stir,ni,n1,n2,n3,stir2,i,j,tmp ; a := 0 ; if n = 0 then RETURN(1) ; fi ; stir := combinat[partition](n) ; stir2 := {} ; for i in stir do if nops(i) <= 3 then tmp := i ; while nops(tmp) < 3 do tmp := [op(tmp),0] ; od: tmp := combinat[permute](tmp) ; for j in tmp do stir2 := stir2 union { j } ; od: fi ; od: for ni in stir2 do n1 := op(1,ni) ; n2 := op(2,ni) ; n3 := op(3,ni) ; a := a+combinat[multinomial](n,n1,n2,n3)*(A000629(3*n1+2*n2+n3)-1/2-2^(3*n1+2*n2+n3)/4)*(-3)^n2*2^n3 ; od: a/(2*6^n) ; end: A136246 := proc(n) local k ; add((-1)^(n-k)*combinat[stirling1](n,k)*A062208(k),k=0..n)/n! ; end: seq(A136246(n),n=0..14) ; # R. J. Mathar, Apr 01 2008

a[n_] := Sum[Binomial[Binomial[j, 3] + n - 1, n] * Sum[(-1)^(i - j)* Binomial[i, j], {i, j, 3n}], {j, 0, 3n}];
a /@ Range[0, 14] (* Jean-François Alcover, Feb 13 2020, after Andrew Howroyd *)

a(n) = {sum(j=0, 3*n, binomial(binomial(j,3)+n-1, n) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

a(n) = Sum_{m>=0} binomial(binomial(m,3)+n-1,n)/2^(m+1).

a(n) = Sum_{j=0..3*n} binomial(binomial(j,3)+n-1, n) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

More terms from R. J. Mathar, Apr 01 2008

Terms a(13) and beyond from Andrew Howroyd, Feb 09 2020

cp's OEIS Frontend

A331567 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column.

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A331510 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of distinct nonzero rows with n ones in every column up to permutation of rows and columns.

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A331278 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

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A331509 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set-systems on a k-set.

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A226994 Number of lattice paths from (0,0) to (n,n) consisting of steps U=(1,1), H=(1,0) and S=(0,1) such that the first step leaving the diagonal (if any) is an H step.

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A331277 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and columns in decreasing lexicographic order.

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A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

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A121316 Unlabeled version of A055203: number of different relations between n intervals (of nonzero length) on a line, up to permutation of intervals.

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A136246 a(n) = (1/n!)Sum_{k=0..n} (-1)^(n-k)Stirling1(n,k)*A062208(k).