A318805 Array read by antidiagonals: T(n,k) is the number of inequivalent symmetric nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 6, 8, 4, 2, 1, 1, 8, 13, 9, 4, 2, 1, 1, 10, 22, 16, 9, 4, 2, 1, 1, 13, 33, 32, 17, 9, 4, 2, 1, 1, 15, 52, 57, 35, 17, 9, 4, 2, 1, 1, 18, 76, 105, 68, 36, 17, 9, 4, 2, 1, 1, 21, 108, 178, 139, 71, 36, 17, 9, 4, 2, 1

Offset

1,5

Links

Table of n, a(n) for n=1..78.

Formula

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

Example

Array begins:
===============================================
n\k| 1 2 3 4  5  6  7   8   9  10   11   12
---+-------------------------------------------
1  | 1 1 1 1  1  1  1   1   1   1    1    1 ...
2  | 1 2 3 5  6  8 10  13  15  18   21   25 ...
3  | 1 2 4 8 13 22 33  52  76 108  150  209 ...
4  | 1 2 4 9 16 32 57 105 178 301  490  793 ...
5  | 1 2 4 9 17 35 68 139 264 502  924 1695 ...
6  | 1 2 4 9 17 36 71 151 303 619 1234 2473 ...
7  | 1 2 4 9 17 36 72 154 315 661 1370 2885 ...
8  | 1 2 4 9 17 36 72 155 318 673 1413 3034 ...
9  | 1 2 4 9 17 36 72 155 319 676 1425 3078 ...
...

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, k_] := SeriesCoefficient[1/(Product[Product[(1 - x^(2*LCM[p[[i]], p[[j]] ]))^GCD[p[[i]], p[[j]]], {j, 1, i - 1}], {i, 2, Length[p]}]* Product[t = p[[i]]; (1 - x^t)^Mod[t, 2]*(1 - x^(2*t))^Quotient[t, 2], {i, 1, Length[p]}]), {x, 0, k}];
T[_, 1] = T[1, _] = 1; T[n_, k_] := (s = 0; Do[s += permcount[p]*c[p, k], {p, IntegerPartitions[n]}]; s/n!);
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 13 2018, 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, k)={polcoef(1/(prod(i=2, #p, prod(j=1, i-1, (1 - x^(2*lcm(p[i], p[j])) + O(x*x^k))^gcd(p[i], p[j]))) * prod(i=1, #p, my(t=p[i]); (1 - x^t + O(x*x^k))^(t%2)*(1 - x^(2*t) + O(x*x^k))^(t\2) )), k)}
T(n, k)={if(n==0, k==0, my(s=0); forpart(p=n, s+=permcount(p)*c(p, k)); s/n!)}

Crossrefs

Cf. A318795.

Main diagonal is A316983.

Sequence in context: A104762 A152462 A180360 * A175331 A098805 A168396

Adjacent sequences: A318802 A318803 A318804 * A318806 A318807 A318808

Keyword

nonn,tabl

Author

Andrew Howroyd, Sep 03 2018

Status

approved