A297388 -id:A297388 - cp's OEIS frontend

A369910 Number of pairs (p,q) of partitions of n such that the set of parts in q is a proper subset of the set of parts in p.

0, 0, 0, 1, 3, 4, 15, 20, 52, 83, 163, 246, 501, 727, 1295, 1994, 3375, 4969, 8267, 12036, 19287, 28270, 43511, 62799, 96364, 137358, 204388, 291607, 427446, 601257, 874088, 1218524, 1743989, 2424096, 3422084, 4718626, 6622937, 9053800, 12559895, 17112883

Offset: 0

Alois P. Heinz, Feb 05 2024

nonn

			a(5) = 4: (2111, 11111), (221, 11111), (311, 11111), (41, 11111).
a(6) = 15: (21111, 111111), (21111, 222), (2211, 111111), (2211, 222), (3111, 111111), (321, 111111), (321, 21111), (321, 2211), (321, 222), (321, 3111), (3111, 33), (321, 33), (411, 111111), (42, 222), (51, 111111).

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A297388, A369704, A369707.

b:= proc(n, m, i, t) option remember; `if`(n=0,
     `if`(t and m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1, t)+add(
      add(b(n-i*j, m-i*h, i-1, h=0 or t), h=0..m/i), j=1..n/i)))
    end:
a:= n-> b(n$3, false):
seq(a(n), n=0..42);

b[n_, m_, i_, t_] := b[n, m, i, t] = If[n == 0,
   If[t && m == 0, 1, 0], If[i < 1, 0, b[n, m, i-1, t] +
   Sum[Sum[b[n-i*j, m-i*h, i-1, h == 0 || t], {h, 0, m/i}], {j, 1, n/i}]]];
a[n_] := b[n, n, n, False];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

A299967 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions non-singleton skew-partitions.

Original entry on oeis.org

1, 0, 2, 3, 13, 32, 121, 376, 1406, 5030, 19632, 76334, 314582, 1308550, 5667494, 24940458, 113239394, 523149560, 2480434938, 11968944532, 59051754824

Offset: 0

Gus Wiseman, Feb 22 2018

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

			The a(4) = 13 tableaux:
1 1 2 2   1 1 1 1
.
1 2 2   1 1 2   1 1 1
1       2       1
.
1 2   1 1   1 1
1 2   2 2   1 1
.
1 2  1 1   1 1
1    2     1
2    2     1
.
1   1
1   1
2   1
2   1

Cf. A000085, A063834, A138178, A153452, A238690, A296188, A297388, A299925, A299926, A299966.

undptns[y_]:=DeleteCases[Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&],0,{2}];
ehn[y_]:=ehn[y]=If[Total[y]=!=1,1,0]+Sum[ehn[c],{c,Select[undptns[y],Total[#]>1&&Total[y]-Total[#]>1&]}];
Table[Sum[ehn[y],{y,IntegerPartitions[n]}],{n,15}]

A305023 Number of triples (p,q,r) of partitions such that p is a partition of n and r <= q <= p (by diagram containment).

Original entry on oeis.org

1, 3, 12, 34, 100, 246, 630, 1433, 3298, 7124, 15283, 31358, 64100, 126406, 247587, 472864, 895548, 1661690, 3059734, 5538991, 9950980, 17631398, 31004004, 53878023, 92979904, 158806852, 269448833, 453099946, 757152246, 1255180557, 2068707378, 3385065586

Offset: 0

Alois P. Heinz, May 23 2018

nonn

			a(0) = 1: ((),(),()).
a(1) = 3: (1,(),()), (1,1,()), (1,1,1).
a(2) = 12: (11,(),()), (11,1,()), (11,1,1), (11,11,()), (11,11,1), (11,11,11), (2,(),()), (2,1,()), (2,1,1), (2,2,()), (2,2,1), (2,2,2).
a(3) = 34: (111,(),()), (111,1,()), (111,1,1), (111,11,()), (111,11,1), (111,11,11), (111,111,()), (111,111,1), (111,111,11), (111,111,111), (21,(),()), (21,1,()), (21,1,1), (21,11,()), (21,11,1), (21,11,11), (21,2,()), (21,2,1), (21,2,2), (21,21,()), (21,21,1), (21,21,11), (21,21,2), (21,21,21), (3,(),()), (3,1,()), (3,1,1), (3,2,()), (3,2,1), (3,2,2), (3,3,()), (3,3,1), (3,3,2), (3,3,3).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram

Cf. A000041, A297388.

cp's OEIS Frontend

A369910 Number of pairs (p,q) of partitions of n such that the set of parts in q is a proper subset of the set of parts in p.

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A299967 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions non-singleton skew-partitions.

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Examples

Crossrefs

Programs

Mathematica

A305023 Number of triples (p,q,r) of partitions such that p is a partition of n and r <= q <= p (by diagram containment).

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Author

Keywords

Examples

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