A209936 Triangle of multiplicities of k-th partition of n corresponding to sequence A080577. Multiplicity of a given partition of n into k parts is the number of ways parts can be selected from k distinguishable bins. See the example.

1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 60, 15, 120, 60, 20, 90, 30, 1, 7, 42, 42, 105, 42, 210, 140, 105, 105, 420, 105, 140, 210, 42, 1, 8, 56, 56, 168, 56, 336, 280, 28, 336, 168, 840, 280, 168, 420, 840, 1120, 168, 70, 560, 420, 56, 1

Offset

1,2

Comments

Differs from A035206 after position 21.

Differs from A210238 after position 21.

The n-th row of the triangle, written as a column vector v(n), satisfies K . v(n) = #SSYT(lambda,n) where K is the Kostka matrix of order n, and #SSYT(lambda,n) is the count of semi-standard Young tableaux in n variables of the partitions of n. - Wouter Meeussen, Jan 27 2025

Links

Table of n, a(n) for n=1..66.

Sergei Viznyuk, C Program

Example

Triangle begins:
  1
  2, 1
  3, 6, 1
  4, 12, 6, 12, 1
  5, 20, 20, 30, 30, 20, 1
  6, 30, 30, 60, 15, 120, 60, 20, 90, 30, 1
  7, 42, 42, 105, 42, 210, 140, 105, 105, 420, 105, 140, 210, 42, 1
  ...
Thus for n=3 (third row) the partitions of n=3 are:
  3+0+0  0+3+0  0+0+3   (multiplicity=3),
  2+1+0  2+0+1  1+2+0  1+0+2  0+2+1  0+1+2  (multiplicity=6),
  1+1+1  (multiplicity=1).

Mathematica

Apply[Multinomial, Last/@Tally[#]&/@PadRight[IntegerPartitions[n]], 1] (* Wouter Meeussen, Jan 26 2025 *)

Crossrefs

Cf. A080577, A078760, A035206, A210238.

Row lengths give A000041.

Row sums give A088218.

Sequence in context: A225632 A035206 A210238 * A213941 A360858 A181511

Adjacent sequences: A209933 A209934 A209935 * A209937 A209938 A209939

Keyword

nonn,tabf

Author

Sergei Viznyuk, Mar 15 2012

Status

approved