A299925 -id:A299925 - cp's OEIS frontend

A299966 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions non-singleton skew-partitions.

1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 3, 3, 5, 5, 5, 2, 8, 5, 13, 6, 13, 10, 21, 5, 11, 18, 11, 14, 34, 15, 55, 3, 26, 33, 23, 13, 89, 59, 54, 14, 144, 38, 233, 28, 31, 105, 377, 10, 47, 31, 106, 57, 610, 23, 60, 32, 206, 185, 987, 38, 1597, 324, 91, 5, 132, 93, 2584, 111

Offset: 1

Gus Wiseman, Feb 22 2018

nonn

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(25) = 11 tableaux:
1 2 3   1 2 2   1 1 3   1 1 2
1 2 3   1 3 3   2 2 3   2 3 3
.
1 2 2   1 1 2   1 1 2   1 1 2   1 1 1   1 1 1
1 2 2   2 2 2   1 2 2   1 1 2   2 2 2   1 2 2
.
1 1 1
1 1 1

Bruce E. Sagan, The Symmetric Group, Springer-Verlag New York, 2001.

Gus Wiseman, The a(30) = 15 non-singleton tableaux of shape (321).

Cf. A000085, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A297388, A299925, A299926, A299967.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
undptns[y_]:=DeleteCases[Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&],0,{2}];
eh[y_]:=If[Total[y]=!=1,1,0]+Sum[eh[c],{c,Select[undptns[y],Total[#]>1&&Total[y]-Total[#]>1&]}];
Table[eh[Reverse[primeMS[n]]],{n,60}]

A285175 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly increasing.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 5, 1, 7, 11, 1, 1, 11, 1, 13, 23, 9, 1, 7, 11, 11, 11, 25, 1, 51, 1, 1, 39, 13, 45, 23, 1, 15, 59, 25, 1, 135, 1, 41, 73, 17, 1, 9, 45, 73, 83, 61, 1, 45, 107, 63, 111, 19, 1, 135, 1, 21, 259, 1, 205, 279, 1, 85, 143, 349, 1

Offset: 1

Gus Wiseman, Feb 26 2018

nonn

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(15) = 11 tableaux:
1 2 3   1 2 4   1 3 4   1 2 5   1 3 5
4 5     3 5     2 5     3 4     2 4
.
1 2 3   1 2 3   1 2 4   1 2 4   1 3 4
2 4     3 4     2 3     3 4     2 4
.
1 2 3
2 3

Cf. A000085, A001221, A005117, A006958, A015128, A056239, A138178, A153452, A238690, A296150, A296188, A297388, A299925, A299926, A299968, A300118, A300120, A300122.

a[n_]:=If[n===1,1,Sum[a[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Array[a,100]

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}]

A318915 Number of joining pairs of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 33, 41, 77, 105, 173, 215, 381, 449, 699, 911, 1335, 1611, 2433, 2867, 4179, 5113, 6903, 8251, 11769, 13661, 18177, 22011, 28997, 33711, 45251

Offset: 0

Gus Wiseman, Sep 05 2018

Two integer partitions are a joining pair if they have no common cover (coarser partition) other than the maximum. For example, (221) and (311) are not a joining pair as they are both covered by (32) or (41), while (222) and (33) are a joining pair.
All terms are odd.
The same as the number of pairs of integer partitions of n without common subsums. - Mamuka Jibladze, Jun 16 2024

			The sequence of joining pairs of integer partitions begins:
  ()()   (1)(1)   (2)(2)    (3)(3)     (4)(4)      (5)(5)
                  (2)(11)   (3)(21)    (4)(31)     (5)(41)
                  (11)(2)   (3)(111)   (4)(22)     (5)(32)
                            (21)(3)    (4)(211)    (5)(311)
                            (111)(3)   (4)(1111)   (5)(221)
                                       (31)(4)     (5)(2111)
                                       (31)(22)    (5)(11111)
                                       (22)(4)     (41)(5)
                                       (22)(31)    (41)(32)
                                       (211)(4)    (32)(5)
                                       (1111)(4)   (32)(41)
                                                   (311)(5)
                                                   (221)(5)
                                                   (2111)(5)
                                                   (11111)(5)

P. Erdős, J. Nicolas and A. Sárközy, On the number of pairs of partitions of n without common subsums, Colloquium Mathematicae, 63 (1992), 61-83.

Cf. A000041, A059849, A060639, A181939, A265947, A299925, A300383, A317141, A317143.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
ptncaps[y_]:=Union[Map[Sort[Total/@#,Greater]&,mps[y],{1}]];
Table[Select[Tuples[IntegerPartitions[n],2],Intersection@@ptncaps/@#=={{n}}&]//Length,{n,6}]

a(n) >= 2 * A000041(n) - 1. - Alois P. Heinz, Sep 06 2018

a(13)-a(30) from Alois P. Heinz, Sep 05 2018

A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 11, 2, 2, 1, 33, 1, 116, 1, 5, 4, 435, 1, 2, 11, 1, 2, 1832, 2, 8167, 1, 12, 33, 10, 1, 39700, 116, 37, 1, 201785, 5, 1099449, 4, 3, 435, 6237505, 1, 19, 2, 123, 11, 37406458, 1, 27, 2, 474, 1832, 232176847, 2, 1513796040

Offset: 1

Gus Wiseman, Mar 04 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(21) = 5 maximal chains are the rows:
(111111)<(21111)<(2211)<(222)<(42)
(111111)<(21111)<(2211)<(411)<(42)
(111111)<(21111)<(2211)<(321)<(42)
(111111)<(21111)<(3111)<(411)<(42)
(111111)<(21111)<(3111)<(321)<(42)

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300385.

pcovs[ptn_]:=Select[Union[Reverse/@Sort/@Join@@@Tuples[IntegerPartitions/@ptn]],Length[#]===Length[ptn]+1&];
coc[ptn_]:=coc[ptn]=If[Max[ptn]===1,1,Total[coc/@pcovs[ptn]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[coc[Reverse[primeMS[n]]],{n,50}]

A300271 Smallest Heinz number of a partition obtained from y by removing one square from its Young diagram, where y is the integer partition with Heinz number n > 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 4, 6, 5, 7, 6, 11, 7, 9, 8, 13, 9, 17, 10, 14, 11, 19, 12, 15, 13, 18, 14, 23, 15, 29, 16, 21, 17, 21, 18, 31, 19, 26, 20, 37, 21, 41, 22, 27, 23, 43, 24, 35, 25, 34, 26, 47, 27, 33, 28, 38, 29, 53, 30, 59, 31, 42, 32, 39, 33, 61, 34, 46

Offset: 2

Gus Wiseman, Mar 01 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Cf. A000041, A000085, A001221, A005117, A056239, A094457, A112798, A122111, A153452, A215366, A238690, A296150, A299925.

dip[n_]:=Min@@Table[n/q*If[q===2,1,NextPrime[q,-1]],{q,Select[Divisors[n],PrimeQ]}];
Table[dip[n],{n,2,50}]

cp's OEIS Frontend

A299966 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions non-singleton skew-partitions.

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A285175 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly increasing.

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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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A318915 Number of joining pairs of integer partitions of n.

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A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

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Mathematica

A300271 Smallest Heinz number of a partition obtained from y by removing one square from its Young diagram, where y is the integer partition with Heinz number n > 1.

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