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A321609 Array read by antidiagonals: T(n,k) is the number of inequivalent binary n X n matrices with k ones, under row and column permutations. 4
1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 1, 6, 3, 1, 1, 0, 0, 0, 7, 6, 3, 1, 1, 0, 0, 0, 7, 16, 6, 3, 1, 1, 0, 0, 0, 6, 21, 16, 6, 3, 1, 1, 0, 0, 0, 3, 39, 34, 16, 6, 3, 1, 1, 0, 0, 0, 1, 44, 69, 34, 16, 6, 3, 1, 1, 0, 0, 0, 1, 55, 130, 90, 34, 16, 6, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

LINKS

Table of n, a(n) for n=0..90.

FORMULA

T(n,k) = T(k,k) for n > k.

T(n,k) = 0 for k > n^2.

EXAMPLE

Array begins:

==========================================================

n\k| 0  1  2  3  4  5  6   7   8    9   10    11    12

---+------------------------------------------------------

0  | 1  0  0  0  0  0  0   0   0    0    0     0     0 ...

1  | 1  1  0  0  0  0  0   0   0    0    0     0     0 ...

2  | 1  1  3  1  1  0  0   0   0    0    0     0     0 ...

3  | 1  1  3  6  7  7  6   3   1    1    0     0     0 ...

4  | 1  1  3  6 16 21 39  44  55   44   39    21    16 ...

5  | 1  1  3  6 16 34 69 130 234  367  527   669   755 ...

6  | 1  1  3  6 16 34 90 182 425  870 1799  3323  5973 ...

7  | 1  1  3  6 16 34 90 211 515 1229 2960  6893 15753 ...

8  | 1  1  3  6 16 34 90 211 558 1371 3601  9209 24110 ...

9  | 1  1  3  6 16 34 90 211 558 1430 3825 10278 28427 ...

...

MATHEMATICA

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];

c[p_List, q_List, k_] := SeriesCoefficient[Product[Product[(1 + O[x]^(k + 1) + x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}], {i, 1, Length[p]}], {x, 0, k}];

M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)]

Table[M[n - k, n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Sep 10 2019, after Andrew Howroyd *)

PROG

(PARI)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

c(p, q, k)={polcoef(prod(i=1, #p, prod(j=1, #q, (1 + x^lcm(p[i], q[j]) + O(x*x^k))^gcd(p[i], q[j]))), k)}

M(m, n, k)={my(s=0); forpart(p=m, forpart(q=n, s+=permcount(p) * permcount(q) * c(p, q, k))); s/(m!*n!)}

for(n=0, 10, for(k=0, 12, print1(M(n, n, k), ", ")); print); \\ Andrew Howroyd, Nov 14 2018

CROSSREFS

Rows n=6..8 are A052370, A053304, A053305.

Main diagonal is A049311.

Row sums are A002724.

Cf. A052371 (as triangle), A057150, A246106, A318795.

Sequence in context: A204181 A204242 A211313 * A238414 A195151 A271024

Adjacent sequences:  A321606 A321607 A321608 * A321610 A321611 A321612

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Nov 14 2018

STATUS

approved

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Last modified April 11 03:12 EDT 2021. Contains 342886 sequences. (Running on oeis4.)