A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Author

Tilman Piesk, Nov 04 2014

Keywords

• nonn
• tabf

Comments

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.

Row lengths: A000041 (partition numbers), Row sums: A005651

Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388

Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012

The rightmost columns form the triangle A173333:

n 0 1 2 4 6 10 14 21 (A000041(1,2,3...)-1)

m

1 1

2 2 1

3 6 3 1

4 24 12 4 1

5 120 60 20 5 1

6 720 360 120 30 6 1

7 5040 2520 840 210 42 7 1

8 40320 20160 6720 1680 336 56 8 1

A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

Examples

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1
		

Links

• Tilman Piesk, Triangle rows m=0..25, flattened
• Tilman Piesk, Illustration of the multisets for m=0..6

Crossrefs

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.