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

Author

Sergei Viznyuk, Mar 15 2012

Keywords

• nonn
• tabf

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

Examples

			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).
		

Links

• Sergei Viznyuk, C Program

Crossrefs

Cf. A080577, A078760, A035206, A210238.

Row lengths give A000041.

Row sums give A088218.

Programs

Mathematica

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