A108470 Table read by antidiagonals: T(n,k) = number of labeled partitions of (n,k) into pairs (i,j).

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 15, 1, 1, 31, 79, 79, 31, 1, 1, 63, 241, 339, 241, 63, 1, 1, 127, 727, 1351, 1351, 727, 127, 1, 1, 255, 2185, 5235, 6721, 5235, 2185, 255, 1, 1, 511, 6559, 20119, 31831, 31831, 20119, 6559, 511, 1, 1, 1023, 19681, 77379

Offset: 1

Author

Christian G. Bower, Jun 03 2005

Keywords

• nonn
• tabl

Comments

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object.

Examples

			1 1 1 1 1 ...
1 3 7 15 31 ...
1 7 25 79 241 ...
1 15 79 339 1351 ...
1 31 241 1351 6721 ...
		

Crossrefs

Cf. A108461. Columns 1-3: A000012, A000225, A058481. Main diagonal: A023997.

Programs

Maxima

T(n,k):=sum(m!*stirling2(k,m)*stirling2(n-k+1,m),m,1,min(k,n-k+1)); /* Vladimir Kruchinin, Apr 11 2015 */

PARI

antidiag(nn) = {for (n=1, nn, for (k=1, n, print1(sum(m=1, min(k, n-k+1), m!*stirling(k, m, 2)*stirling(n-k+1, m, 2)), ", "); ); print(););} \\ Michel Marcus, Apr 11 2015

PARI

tabl(nn) = {default(seriesprecision, nn); for (n=1, nn, for (k=1, nn, print1(k!*polcoeff(polcoeff(n!*exp((exp(x)-1)*(exp(y)-1))+O(x^(n+1)), n, x), k, y), ", "); ); print(););} \\ Michel Marcus, Apr 11 2015

Formula

Double e.g.f.: exp((exp(x)-1)*(exp(y)-1)).

T(n,k) = Sum{m=1..min(k,n-k+1)} m!*stirling2(k,m)*stirling2(n-k+1,m). - Vladimir Kruchinin, Apr 11 2015