Combinatorial Interpretation of General Eulerian Numbers

Table 1

The map 𝑓∶{𝐴𝑊_{𝑛−1,𝑘},𝐴𝑊_{𝑛−1,𝑘−1}}→𝐴𝑊_𝑛,𝑘.

𝐴=𝑝1𝑝2…𝑝𝑛−1	Position 𝑖	Condition	𝐴^′∈𝐴𝑊𝑛,𝑘
𝐴∈𝐴𝑊𝑛−1,𝑘	1<𝑖≤𝑛−1; and; 𝑝𝑖≥𝑖	𝑝𝑖>𝑝1	𝐴^′=𝑝1𝑝2…𝑝𝑖−1𝑛𝑝𝑖+1…𝑝𝑛−1𝑝𝑖; With 𝑝1<𝑝𝑛−1,𝑝1<𝑝𝑖; Case  (a1)
		𝑝𝑖<𝑝1  and; 𝑝𝑛−1<𝑛−1	𝐴^′=𝑝1𝑝2…𝑝𝑖−1𝑛𝑝𝑖+1…𝑝𝑛−2𝑝𝑖𝑝𝑛−1; With 𝑝𝑖< 𝑝1<𝑝𝑛−1<𝑛−1; Case  (a3)
		𝑝𝑖<𝑝1  and; 𝑝𝑛−1=𝑛−1	𝐴^′=𝑝1𝑝2…𝑝𝑖−1𝑛−1𝑝𝑖+1…𝑝𝑛−2𝑝𝑖𝑛; With 𝑝𝑖< 𝑝1,𝑝𝑖 ≥𝑖; Case  (b2)
	𝑖=1	𝑝𝑗=𝑛−1; and  𝑗<𝑛−1	𝐴^′=𝑝𝑛−1𝑝2…𝑝𝑗−1𝑛𝑝𝑗+1…𝑝𝑛−2𝑝1𝑛−1; With 𝑝1<𝑝𝑛−1; Case  (a5)
		𝑝𝑛−1=𝑛−1	𝐴^′=𝑛−1𝑝2…𝑝𝑛−2𝑝1𝑛; Case  (b4)
𝐴∈𝐴𝑊𝑛−1,𝑘−1	1<𝑖≤𝑛−1; and; 𝑝𝑖<𝑖	𝑝𝑖>𝑝1	𝐴^′=𝑝1𝑝2…𝑝𝑖−1𝑛𝑝𝑖+1…𝑝𝑛−1𝑝𝑖; With 𝑝1<𝑝𝑛−1,𝑝1<𝑝𝑖; Case  (a2)
		𝑝𝑖<𝑝1  and; 𝑝𝑛−1<𝑛−1	𝐴^′=𝑝1𝑝2…𝑝𝑖−1𝑛𝑝𝑖+1…𝑝𝑛−2𝑝𝑖𝑝𝑛−1; With 𝑝𝑖< 𝑝1<𝑝𝑛−1<𝑛−1; Case  (a4)
		𝑝𝑖<𝑝1  and; 𝑝𝑛−1=𝑛−1	𝐴^′=𝑝1𝑝2…𝑝𝑖−1𝑛−1𝑝𝑖+1…𝑝𝑛−2𝑝𝑖𝑛; With 𝑝𝑖< 𝑝1,𝑝𝑖 <𝑖; Case  (b3)
	𝑖=𝑛		𝐴^′=𝑝1𝑝2…𝑝𝑛−1𝑛; Case  (b1)