- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤ

Values

[1] => [1] => 1

[1,2] => [2] => 1

[2,1] => [1,1] => 1

[1,2,3] => [3] => 1

[1,3,2] => [2,1] => 2

[2,1,3] => [1,2] => 2

[2,3,1] => [2,1] => 2

[3,1,2] => [1,2] => 2

[3,2,1] => [1,1,1] => 1

[1,2,3,4] => [4] => 1

[1,2,4,3] => [3,1] => 3

[1,3,2,4] => [2,2] => 5

[1,3,4,2] => [3,1] => 3

[1,4,2,3] => [2,2] => 5

[1,4,3,2] => [2,1,1] => 3

[2,1,3,4] => [1,3] => 3

[2,1,4,3] => [1,2,1] => 5

[2,3,1,4] => [2,2] => 5

[2,3,4,1] => [3,1] => 3

[2,4,1,3] => [2,2] => 5

[2,4,3,1] => [2,1,1] => 3

[3,1,2,4] => [1,3] => 3

[3,1,4,2] => [1,2,1] => 5

[3,2,1,4] => [1,1,2] => 3

[3,2,4,1] => [1,2,1] => 5

[3,4,1,2] => [2,2] => 5

[3,4,2,1] => [2,1,1] => 3

[4,1,2,3] => [1,3] => 3

[4,1,3,2] => [1,2,1] => 5

[4,2,1,3] => [1,1,2] => 3

[4,2,3,1] => [1,2,1] => 5

[4,3,1,2] => [1,1,2] => 3

[4,3,2,1] => [1,1,1,1] => 1

[1,2,3,4,5] => [5] => 1

[1,2,3,5,4] => [4,1] => 4

[1,2,4,3,5] => [3,2] => 9

[1,2,4,5,3] => [4,1] => 4

[1,2,5,3,4] => [3,2] => 9

[1,2,5,4,3] => [3,1,1] => 6

[1,3,2,4,5] => [2,3] => 9

[1,3,2,5,4] => [2,2,1] => 16

[1,3,4,2,5] => [3,2] => 9

[1,3,4,5,2] => [4,1] => 4

[1,3,5,2,4] => [3,2] => 9

[1,3,5,4,2] => [3,1,1] => 6

[1,4,2,3,5] => [2,3] => 9

[1,4,2,5,3] => [2,2,1] => 16

[1,4,3,2,5] => [2,1,2] => 11

[1,4,3,5,2] => [2,2,1] => 16

[1,4,5,2,3] => [3,2] => 9

[1,4,5,3,2] => [3,1,1] => 6

[1,5,2,3,4] => [2,3] => 9

[1,5,2,4,3] => [2,2,1] => 16

[1,5,3,2,4] => [2,1,2] => 11

[1,5,3,4,2] => [2,2,1] => 16

[1,5,4,2,3] => [2,1,2] => 11

[1,5,4,3,2] => [2,1,1,1] => 4

[2,1,3,4,5] => [1,4] => 4

[2,1,3,5,4] => [1,3,1] => 11

[2,1,4,3,5] => [1,2,2] => 16

[2,1,4,5,3] => [1,3,1] => 11

[2,1,5,3,4] => [1,2,2] => 16

[2,1,5,4,3] => [1,2,1,1] => 9

[2,3,1,4,5] => [2,3] => 9

[2,3,1,5,4] => [2,2,1] => 16

[2,3,4,1,5] => [3,2] => 9

[2,3,4,5,1] => [4,1] => 4

[2,3,5,1,4] => [3,2] => 9

[2,3,5,4,1] => [3,1,1] => 6

[2,4,1,3,5] => [2,3] => 9

[2,4,1,5,3] => [2,2,1] => 16

[2,4,3,1,5] => [2,1,2] => 11

[2,4,3,5,1] => [2,2,1] => 16

[2,4,5,1,3] => [3,2] => 9

[2,4,5,3,1] => [3,1,1] => 6

[2,5,1,3,4] => [2,3] => 9

[2,5,1,4,3] => [2,2,1] => 16

[2,5,3,1,4] => [2,1,2] => 11

[2,5,3,4,1] => [2,2,1] => 16

[2,5,4,1,3] => [2,1,2] => 11

[2,5,4,3,1] => [2,1,1,1] => 4

[3,1,2,4,5] => [1,4] => 4

[3,1,2,5,4] => [1,3,1] => 11

[3,1,4,2,5] => [1,2,2] => 16

[3,1,4,5,2] => [1,3,1] => 11

[3,1,5,2,4] => [1,2,2] => 16

[3,1,5,4,2] => [1,2,1,1] => 9

[3,2,1,4,5] => [1,1,3] => 6

[3,2,1,5,4] => [1,1,2,1] => 9

[3,2,4,1,5] => [1,2,2] => 16

[3,2,4,5,1] => [1,3,1] => 11

[3,2,5,1,4] => [1,2,2] => 16

[3,2,5,4,1] => [1,2,1,1] => 9

[3,4,1,2,5] => [2,3] => 9

[3,4,1,5,2] => [2,2,1] => 16

[3,4,2,1,5] => [2,1,2] => 11

[3,4,2,5,1] => [2,2,1] => 16

[3,4,5,1,2] => [3,2] => 9

[3,4,5,2,1] => [3,1,1] => 6

[3,5,1,2,4] => [2,3] => 9

[3,5,1,4,2] => [2,2,1] => 16

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 2\ q + 4\ q^{2}$

$F_{4} = 2\ q + 12\ q^{3} + 10\ q^{5}$

$F_{5} = 2\ q + 16\ q^{4} + 12\ q^{6} + 36\ q^{9} + 22\ q^{11} + 32\ q^{16}$

$F_{6} = 2\ q + 20\ q^{5} + 40\ q^{10} + 56\ q^{14} + 76\ q^{19} + 104\ q^{26} + 140\ q^{35} + 160\ q^{40} + 122\ q^{61}$

$F_{7} = 2\ q + 24\ q^{6} + 60\ q^{15} + 120\ q^{20} + 58\ q^{29} + 136\ q^{34} + 200\ q^{50} + 220\ q^{55} + 256\ q^{64} + 142\ q^{71} + 312\ q^{78} + 180\ q^{90} + 396\ q^{99} + 444\ q^{111} + 264\ q^{132} + 620\ q^{155} + 338\ q^{169} + 724\ q^{181} + 544\ q^{272}$

Description

The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a

$2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition

$c=c_1,\dots,c_k$ of

$n$ , the number of ribbon shaped standard tableaux equals

$\sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell},$
where the sum is over all coarsenings of

$c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation

$\pi$ of length

$n$ is the integer composition of

$n$ whose descent set equals the descent set of

$\pi$ . The descent set of a permutation

$\pi$ is

$\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$ . The descent set of a composition

$c = (i_1, i_2, \ldots, i_k)$ is the set

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .