A238690 -id:A238690 - cp's OEIS frontend

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

1, 4, 13, 51, 183, 771, 3087, 13601, 59933, 278797, 1311719, 6453606, 32179898, 166075956, 871713213, 4704669005, 25831172649, 145260890323

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. 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(3) = 13 tableaux:
1 1 1   1 1 2   1 2 2   1 2 3
.
1 1   1 1   1 2   1 2   1 3
1     2     1     3     2
.
1   1   1   1
1   1   2   2
1   2   2   3

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A296561, A297388, A299699, A299925, A299926, A300118, A300120, A300121, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A300123 Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 4, 10, 8, 16, 8, 32, 16, 20, 8, 64, 20, 128, 16, 40, 32, 256, 16, 52, 64, 52, 32, 512, 40, 1024, 16, 80, 128, 104, 40, 2048, 256, 160, 32, 4096, 80, 8192, 64, 104, 512, 16384, 32, 272, 104

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Gus Wiseman, The a(25) = 52 connected tilings of (3,3).

Cf. A000085, A000898, A056239, A006958, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300060, A300118, A300120, A300121, A300122, A300124.

A300118 Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 6, 5, 6, 5, 7, 6, 7, 5, 8, 7, 9, 6, 8, 7, 10, 6, 10, 8, 10, 7, 11, 8, 12, 6, 9, 9, 11, 8, 13, 10, 10, 7, 14, 9, 15, 8, 11, 11, 16, 7, 15, 11, 11, 9, 17, 11, 12, 8, 12, 12, 18, 9, 19, 13, 12, 7, 13, 10, 20, 10, 13, 12, 21, 9, 22, 14, 15, 11

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

			The a(15) = 7 denominators are (), (1), (11), (22), (3), (31), (32) with diagrams:
o o o   . o o   . o o   . . o   . . .   . . .   o o o
o o     o o     . o     . .     o o     . o     o o
Missing are the two disconnected skew partitions:
. . o   . . o
o o     . o

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A297388, A299925, A299926, A300056, A300060, A300120, A300122, A300123, A300124.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

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.

Original entry on oeis.org

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]

A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 7, 4, 6, 6, 9, 7, 7, 9, 11, 10, 8, 12, 9, 13, 5, 10, 13, 9, 15, 14, 15, 9, 14, 16, 10, 18, 19, 17, 13, 18, 10, 19, 11, 16, 21, 12, 15, 24, 19, 17, 22, 16, 22, 12, 23, 24, 6, 19, 20, 29, 21, 21, 26, 22, 25, 13, 30, 27, 11, 26, 25, 19, 34

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Also the number of members of A025487 that divide A025487(n).

			5 members of A025487 divide A025487(6) = 12 (namely, 1, 2, 4, 6 and 12); therefore, a(6) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A085082, A025487, A181822, A322584.
Rearrangement of A115728, A115729 and A238690.
A116473(n) is the number of times n appears in the sequence.

lpsQ[n_] := n == 1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]); lps = Select[Range[6000], lpsQ]; c[n_] := Count[Divisors[n], ?(MemberQ[lps, #] &)]; c /@ lps  (* _Amiram Eldar, Jan 21 2024 *)

a(n) = A085082(A025487(n)) = A085082(A181822(n)).

a(n) = A322584(A025487(n)). - Amiram Eldar, Jan 21 2024

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}]

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

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

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A300123 Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.

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A300118 Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.

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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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A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

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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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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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Mathematica