Enumeration formulas for standard young tableaux of approximate C shape

This paper focuses on the combinatorial enumeration of the truncated hollow standard Young tableaux of approximate C shape. The enumeration formula of approximately C-shaped SYT, namely the convolution of Catalan numbers, is derived by using the multi-integral method based on the sequence statistical model. Meanwhile, it is found that the first and last rows differ in the number of boxes. In the case where the number of boxes in the last row n is large enough, the number of SYT does not change, especially if m = 1, whether it is closely related to the Catalan number. In addition, we present two main closed formulas for SYTs of angular and “T” shapes.


Introduction
The Enumeration for the standard Young tableaux (SYT) is an important research object in enumerative combinatorics. The number of truncated hollow SYT has been studied recently. G. D. James and M. H. Peel introduced the truncated SYT in [1]. In [2], a new explanation was given for truncated SYTs. R.M. Adin and Y. Roichman discussed about the number of SYTs deleted from some cells in the northeast corner in the literature. The work of Adin et al. uses pivoting theory method in [3] to obtain the number of rectangles and moving staircase truncated by a square or approximate square. In [4], G. Panova presented the formula of a rectangle truncated by staircase or by approximate square according to Schur function. P. Sun pointed out in [5,6] that the number of SYTs of regular, shifted and truncated shapes can be derived by evaluating the distribution of nested sequence statistics. A partition  of a positive integer n is a non-increasing sequence of nonnegative integers ( ) 1  2 cells which belong to  rather than  and the shape /  is the cells which belong to  rather than  when cells in the upper left corners are kicked out [8].
The SYT of a hollow truncated shape is denoted by 00 \ |{( , )} ij  that the diagram of shape  deletes the cells belonging to shape  from  is filled by a corresponding truncated diagram with the integers from 1 to | | | |  − so that each row and column is increasing in [4].
For example, 3 ( 1) \ ( ) |{(2,2)} kk + is denoted as Figure 1. Many scholars have studied enumeration of hollow truncated SYTs. And in [9], P.Sun obtained the formulas for SYT of nearly hollow rectangular shapes and gave the definition of the hollow SYT. In addition, the SYT of shapes like number, letter or other shapes are attracting more and more attention. Lu Chen and Chuanjuan Sun gave the number of 4-and H-shaped SYTs in [10]. As a continuation of the work in this paper, we have obtained the enumeration formulas of the following three near-Cshaped hollow truncated SYTs in Figure 2, especially (b), which is related to the Catalan number. This paper is structured as follows. In Section 1, we introduce the previous work of the enumeration of SYT and illustrate the main definitions about SYT of different shapes. Meanwhile, we present the main structure and contribution of this paper. In Section 2, we present the sum of enumeration formula of hollow truncated SYT of approximate C shape: ( ,1 , ),(1 , 1) m k n n k m    , and our method is multiple integration based on the sequence statistical model of SYT in [5]. In Section 3, we obtain the main inferences: (i) In counting the number of SYT is Catalan number in [11, A000108].
(iv) If =1 n , we get the number of the SYT of angular shape, it is a simple combination. (v) As the especial one of (iv), we also easily get the number of T-shaped SYT.

Sum of a combination
This paper focuses on the representation formula of SYT of approximate C shape by means of multiple integration in the nested sequence statistical model combined with the probability distribution 3 function of uniform distribution in the probability theory proposed by P Sun in [5]. And we also use combinatorial identities to prove the formula. Proposition 1. [5] let ( ) Such that: And the joint probability distribution function of the nested sequence statistics composed of (1) where S  is its corresponding nested simplex, which can be expressed as: And for each sample point  in the following event A, And each event corresponds to a SYT diagram of shape  . Since the samples of the event are independent and identically distributed, all the sample points of the event A are possible. From a discrete point of view, the distribution of (0,1) U nested sequence statistics is calculated as Proposition 2. Proposition 2. [5] For the  -type nested sequence statistic that obeys the (0,1) U distribution, its distribution is:  (2) where N  is the number of SYT-type chart of shape  , and 12 || d     = + + + . Proposition 3. [5] Let N  be the number of SYT diagram of shape  , then    11 . 12 m kn n m k n m k H kn Moreover, by changing the variables