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

1,2

Comments

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

Links

Alois P. Heinz, Rows n = 1..50, flattened

M. F. Hasler, A092288, rows 1 - 18.

Example

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

Mathematica

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

Prog

(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(m<n, for(i=1, #P, t=A092288(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

Crossrefs

Column k=1 gives A090539.

Row sums give A319648.

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

Cf. A000219, A066633.

Sequence in context: A230534 A177822 A091156 * A111964 A242351 A124324

Adjacent sequences: A092285 A092286 A092287 * A092289 A092290 A092291

Keyword

nonn,tabl

Author

Wouter Meeussen, Feb 01 2004

Status

approved