A090539 -id:A090539 - cp's OEIS frontend

A091298 Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.

1, 1, 2, 1, 2, 3, 1, 4, 3, 5, 1, 4, 7, 5, 7, 1, 6, 10, 13, 7, 11, 1, 6, 14, 20, 19, 11, 15, 1, 8, 18, 33, 32, 31, 15, 22, 1, 8, 25, 43, 56, 54, 43, 22, 30, 1, 10, 29, 66, 81, 99, 78, 64, 30, 42, 1, 10, 37, 83, 126, 150, 148, 118, 88, 42, 56, 1, 12, 44, 114, 174, 246, 235, 230, 166, 124, 56, 77

Offset: 1

Wouter Meeussen, Feb 24 2004

First column is 1, representing the single-part {{n}}, last column is P(n), since the all-ones plane partitions form the Ferrers-Young plots of the (linear) partitions of n.

A plane partition of n is a two-dimensional table (or matrix) with nonnegative elements summing up to n, and nonincreasing rows and columns. (Zero rows and columns are ignored.) - M. F. Hasler, Sep 22 2018

			This plane partition of n=7: {{3,1,1},{2}} contains 4 parts: 3,1,1,2.
Triangle T(n,k) begins:
  1;
  1,  2;
  1,  2,  3;
  1,  4,  3,  5;
  1,  4,  7,  5,  7;
  1,  6, 10, 13,  7, 11;
  1,  6, 14, 20, 19, 11, 15;
  1,  8, 18, 33, 32, 31, 15, 22;
  1,  8, 25, 43, 56, 54, 43, 22, 30;
  1, 10, 29, 66, 81, 99, 78, 64, 30, 42;
  ...

Alois P. Heinz, Rows n = 1..50
A. Rovenchak, Enumeration of plane partitions with a restricted number of parts, arXiv preprint arXiv:1401.4367 [math-ph], 2014.
E. W. Weisstein, Plane partition.
Wikipedia, Plane partition.

Row sums give A000219.
Cf. A090539, A092288, A089924.
Column 1 is A000012. Column 2 is A052928. Diagonal and subdiagonal are A000041.

(* see A089924 for "planepartition" *) Table[Length /@ Split[Sort[Length /@ Flatten /@ planepartitions[n]]], {n, 16}]

A091298(n,k)=sum(i=1,#n=PlanePartitions(n),sum(j=1,#n[i],#n[i][j])==k)
PlanePartitions(n,L=0,PP=List())={ n<2&&return([if(n,[[1]],[])]); for(N=1,n, my(P=apply(Vecrev, if(L, select(p->vecmin(L-Vecrev(p,#L))>=0, partitions(N,L[1],#L)), partitions(N)))); if(NM. F. Hasler, Sep 24 2018

A090984 a(n) is the number of pairs (x,y) where x is plane partition of n+1 and y is a plane partition of n and x covers y.

Original entry on oeis.org

1, 3, 9, 21, 48, 102, 213, 421, 819, 1542, 2854, 5172, 9240, 16233, 28182, 48288, 81862, 137295, 228153, 375658, 613554, 994155, 1599309, 2554932, 4055406, 6397160, 10032907, 15647277, 24275455, 37471066, 57562533, 88018488, 133996590, 203126712, 306671525, 461184246, 690935892, 1031379271

Offset: 0

Wouter Meeussen, Feb 28 2004

nonn

x = (x_1_1 .. x_1_u1)(x_2_1 .. x_2_u2) .. (x_k_1 .. x_k_uk) y = (y_1_1 .. y_1_v1)(y_2_1 .. y-2_v2) .. (y_m_1 .. y_m_vm) x covers y iff ui >= vi, k >= m, x_i_j >= y_i_j, or, the 3-dimensional Ferrers plot of y falls within that of x.

The analog for ordinary partitions and 2D-Ferrers plots gives A000070.

Suresh Govindarajan, Table of n, a(n) for n = 0..40

Cf. A000070, A090539.

coversplaneQ[parent_?planepartitionQ, child_?planepartitionQ] := Block[{dif=Length[parent]-Length[child], p=Length/@ parent, c=PadRight[Length/@ child, Length[parent], 0]}, And[dif>=0, Min[p-c]>=0, Min[parent-MapThread[PadRight[ #1, #2, 0]&, { PadRight[child, Length[parent], {{0}}], p}]]>=0]]; Table[Count[Outer[coversplaneQ, planepartitions[k], planepartitions[k-1], 1], True, -1], {k, 12}]

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

Original entry on oeis.org

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

Wouter Meeussen, Feb 01 2004

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

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

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

Column k=1 gives A090539.
Row sums give A319648.
T(2n+1,n+1) gives A091360.
Cf. A000219, A066633.

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

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

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A091298 Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.

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A090984 a(n) is the number of pairs (x,y) where x is plane partition of n+1 and y is a plane partition of n and x covers y.

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

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