A092288 Triangle read by rows: T(n,k) = count of parts k in all plane partitions of n.

1, 4, 1, 11, 2, 1, 28, 7, 2, 1, 62, 15, 5, 2, 1, 137, 38, 13, 5, 2, 1, 278, 76, 28, 11, 5, 2, 1, 561, 164, 60, 26, 11, 5, 2, 1, 1080, 316, 124, 52, 24, 11, 5, 2, 1, 2051, 623, 244, 108, 50, 24, 11, 5, 2, 1, 3778, 1156, 469, 208, 100, 48, 24, 11, 5, 2, 1, 6885, 2160, 886, 404, 194, 98, 48, 24, 11, 5, 2, 1

Offset: 1

Author

Wouter Meeussen, Feb 01 2004

Keywords

• nonn
• tabl

Comments

For large n the rows end in A091360 = partial sums of A000219 (count of plane partitions).

Examples

			Triangle begins:
    1;
    4,  1;
   11,  2,  1;
   28,  7,  2,  1;
   62, 15,  5,  2,  1;
  137, 38, 13,  5,  2,  1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..50, flattened
• M. F. Hasler, A092288, rows 1 - 18.

Crossrefs

Column k=1 gives A090539.

Row sums give A319648.

T(2n+1,n+1) gives A091360.

Cf. A000219, A066633.

Programs

Mathematica

Table[Length /@ Split[Sort[Flatten[planepartitions[k]]]], {k, 12}]

PARI

A092288_row(n, c=vector(n), m, k)={for(i=1, #n=PlanePartitions(n), for(j=1,#m=n[i], for(i=1,#k=m[j], c[k[i]]++))); c} \\ See A091298 for PlanePartitions(). See below for more efficient code.
M92288=[]; A092288(n,k,L=0)={n>1||return(if(L,[n,n==k],n==k)); if(#L&& #L<3, my(j=setsearch(M92288,[[n,k,L],[]],1)); j<=#M92288&& M92288[j][1]==[n,k,L]&& return(M92288[j][2])); my(c(p)=sum(i=1,#p,p[i]==k),S=[0,0],t); for(m=1,n,my(P=if(L,select(p->vecmin(L-Vecrev(p,#L))>=0, partitions(m,L[1],#L)), partitions(m))); if(mA092288(n-m,k,Vecrev(P[i])); S+=[t[1], t[1]*c(P[i])+t[2]], S+=[#P,vecsum(apply(c,P))])); if(L, #L<3&& M92288= setunion(M92288,[[[n,k,L],S]]);S,S[2])} \\ M. F. Hasler, Sep 26 2018