A103921 -id:A103921 - cp's OEIS frontend

A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

1, 1, 2, 1, 3, 2, 1, 4, 3, 4, 2, 1, 5, 4, 6, 3, 4, 2, 1, 6, 5, 8, 9, 4, 6, 8, 3, 4, 2, 1, 7, 6, 10, 12, 5, 8, 9, 12, 4, 6, 8, 3, 4, 2, 1, 8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1, 9, 8, 14, 18, 20, 7, 12, 15, 16, 20, 24, 27, 6, 10, 12, 16, 18, 24, 5, 8, 9, 12, 16, 4

Offset: 0

Alford Arnold, May 01 2006

Let Theta(n) denote the set of norm values corresponding to all the partitions of n. The following results hold regarding this set: (i) Theta(n) is a subset of Theta(n+1); (ii) A prime p will appear as a norm only for partitions of n>=p; (iii) There exists a prime p not in Theta(n) for all n>=6; (iv) Let h(k) be the prime floor function which gives the greatest prime less than or equal to the k, then the prime p=h(n+1) does not belong to Theta(n); and (v) The primes not in the set Theta(n) are A000720(A000792(n)) - A000720(n). - Abhimanyu Kumar, Nov 25 2020

			a(9) = 4 because the 9th partition is [2,2] and 2*2 = 4.
Table T(n,k) starts:
  1;
  1;
  2, 1;
  3, 2,  1;
  4, 3,  4,  2,  1;
  5, 4,  6,  3,  4, 2,  1;
  6, 5,  8,  9,  4, 6,  8,  3,  4,  2, 1;
  7, 6, 10, 12,  5, 8,  9, 12,  4,  6, 8, 3, 4,  2,  1;
  8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1;

Abramowitz and Stegun, Handbook (1964) page 831.

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Walter Bridges and William Craig, On the distribution of the norm of partitions, arXiv:2308.00123 [math.CO], 2023.
Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).
Wolfdieter Lang, Rows n=1..10.
Robert Schneider and Andrew V. Sills, The Product of Parts or "Norm" of a Partition, INTEGERS, Volume 20A (2020) Paper #A13, 16 pp.
Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019-2021.

Cf. A000041 (row lengths), A006906 (row sums).

Cf. A005361, A036035, A036036, A048996, A078812, A085643, A103921, A111786.

C(sig)={vecprod(sig)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 19 2020

a(n) = A085643(n)/A048996(n).

T(n,k) = A005361(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

Corrected and extended by Franklin T. Adams-Watters, May 26 2006

A335123 Minimum part of the n-th integer partition in Abramowitz-Stegun order (sum/length/lex); a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 7, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 24 2020

			Triangle begins:
  0
  1
  2 1
  3 1 1
  4 2 1 1 1
  5 2 1 1 1 1 1
  6 3 2 1 2 1 1 1 1 1 1
  7 3 2 1 2 1 1 1 1 1 1 1 1 1 1
  8 4 3 2 1 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Partition minima of A334301.
The length of the same partition is A036043.
The Heinz number of the same partition is A334433.
The number of distinct parts in the same partition is A334440.
The maximum of the same partition is A334441.
The version for reversed partitions is A335124.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in (sum/length/revlex) order are A334439.
Cf. A026791, A049085, A103921, A124734, A185974, A193073, A334302, A334433.

Table[If[n==0,{0},Min/@Sort[IntegerPartitions[n]]],{n,0,8}]

a(n) = A055396(A334433(n)).

A335124 Minimum part of the n-th reversed integer partition in Abramowitz-Stegun order; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 1, 1, 1, 1, 6, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 7, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 2, 3, 4, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 9, 1, 2, 3, 4, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 24 2020

The ordering of reversed partitions is first by sum, then by length, and finally lexicographically. The version for non-reversed partitions is A335123.

			Triangle begins:
  0
  1
  2 1
  3 1 1
  4 1 2 1 1
  5 1 2 1 1 1 1
  6 1 2 3 1 1 2 1 1 1 1
  7 1 2 3 1 1 1 2 1 1 1 1 1 1 1
  8 1 2 3 4 1 1 1 2 2 1 1 1 1 2 1 1 1 1 1 1 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Partition minima of A036036.
The length of the same partition is A036043.
The maximum of the same partition is A049085.
The number of distinct parts in the same partition is A103921.
The Heinz number of the same partition is A185974.
The version for non-reversed partitions is A335123.
Lexicographically ordered reversed partitions are A026791.
Partitions in (sum/length/colex) order are A036037.
Partitions in opposite Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A124734, A193073, A334301, A334302, A334433, A334435.

Table[If[n==0,{0},Min/@Sort[Reverse/@IntegerPartitions[n]]],{n,0,8}]

a(n) = A055396(A185974(n)).

A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 3, 2, 4, 1, 6, 4, 2, 5, 1, 3, 2, 1, 7, 4, 3, 5, 2, 6, 1, 4, 2, 1, 8, 5, 3, 6, 2, 7, 1, 4, 3, 1, 5, 2, 1, 9, 5, 4, 6, 3, 7, 2, 8, 1, 4, 3, 2, 5, 3, 1, 6, 2, 1, 10, 6, 4, 7, 3, 8, 2, 9, 1, 5, 3, 2, 5, 4, 1, 6, 3, 1, 7, 2, 1, 4, 3, 2, 1

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(21)
  4: (4)(31)
  5: (5)(32)(41)
  6: (6)(42)(51)(321)
  7: (7)(43)(52)(61)(421)
  8: (8)(53)(62)(71)(431)(521)
  9: (9)(54)(63)(72)(81)(432)(531)(621)

Wikiversity, Lexicographic and colexicographic order

Starting with reversed partitions gives A026793.
The version for compositions is A124734.
Showing partitions as Heinz numbers gives A246867.
The non-strict version is A334301 (reversed: A036036).
Ignoring length gives A344086 (reversed: A246688).
Same as A344089 with partitions reversed.
The version for revlex instead of lex is A344092.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A185974, A193073, A211992, A296774, A334302, A334433, A334435, A334438, A334439, A334440, A334441, A334442, A344091.

Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

A344092 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 4, 1, 3, 2, 6, 5, 1, 4, 2, 3, 2, 1, 7, 6, 1, 5, 2, 4, 3, 4, 2, 1, 8, 7, 1, 6, 2, 5, 3, 5, 2, 1, 4, 3, 1, 9, 8, 1, 7, 2, 6, 3, 5, 4, 6, 2, 1, 5, 3, 1, 4, 3, 2, 10, 9, 1, 8, 2, 7, 3, 6, 4, 7, 2, 1, 6, 3, 1, 5, 4, 1, 5, 3, 2, 4, 3, 2, 1

Offset: 0

Gus Wiseman, May 14 2021

First differs from A118457 at a(53) = 4, A118457(53) = 2.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
   0: ()
   1: (1)
   2: (2)
   3: (3)(21)
   4: (4)(31)
   5: (5)(41)(32)
   6: (6)(51)(42)(321)
   7: (7)(61)(52)(43)(421)
   8: (8)(71)(62)(53)(521)(431)
   9: (9)(81)(72)(63)(54)(621)(531)(432)

Wikiversity, Lexicographic and colexicographic order

Same as A026793 with rows reversed.
Ignoring length gives A118457.
The non-strict version is A334439 (reversed: A036036/A334302).
The version for lex instead of revlex is A344090.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A193073 reads off lexicographically ordered partitions.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A124734, A185974, A211992, A296774, A334301, A334433, A334435, A334438, A334441.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

cp's OEIS Frontend

A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A335123 Minimum part of the n-th integer partition in Abramowitz-Stegun order (sum/length/lex); a(0) = 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A335124 Minimum part of the n-th reversed integer partition in Abramowitz-Stegun order; a(0) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A344092 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, and finally reverse-lexicographically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica